Sunny Kumar

Vijay Kumar Bhat

Metric Dimension of Radially Symmetrical Plane Graph Obtained from Heptagonal Circular Ladder

Keywords: 2010 Mathematics Subject Classification: 05C10, 05C12 Resolving set, Metric basis, Heptagonal circular ladder, Metric Dimension, Convex Polytopes, Independent Resolving set

 and demonstrate that just 3 vertices fittingly chosen are adequate to resolve all the vertices of these classes of plane graphs.

Introduction

Let Φ = Φ(V, E) be a basic associated (i.e., simple and connected) undirected (i.e., all edges are unidirectional, with no direction associated with them) graph (or network) with an edge set E and vertex set V. The metric dimension of the network Φ = Φ(V, E) is the minimal number of nodes (vertices or hubs) in a set with the property that the rundown of good ways from any vertex to those in the set exceptionally distinguishes that vertex. By definition, for an arranged (ordered) subset F = { 1 , 2 , 3 , ..., z } of hubs in a graph Φ and a node of Φ, the metric code/metric representation of concerning F is the ordered z-tuple (or z-vector) ζ( |F) := (d Φ ( , 1 ), d Φ ( , 2 ), ..., d Φ ( , z )). In the event that each pair of unmistakable vertices of Φ have distinctive metric representations then the arranged set F is known as a resolving (locating) set of Φ. The metric dimension or the location number of Φ is the cardinality of the subset F i.e., location number (or metric dimension) of the graph Φ, indicated by dim(Φ) or β(Φ), is the smallest size of locating set on the graph Φ; formally, β(Φ) = dim(Φ) = minimum{|F| : F is locating set (or resolving set)}. It is realized that the issue of registering this invariant is N P -hard. If a subset T of the arrangement of nodes V(Φ) is both resolving and independent (or autonomous), then the set T is known as an independent resolving set (or an autonomous resolving set) for the graph Φ.

For an organized set (or ordered set) of nodes F = { 1 , 2 , 3 , ..., z } of Φ, the pth component (or distance coordinate) of the code ζ( |F) is zero iff = p . Subsequently, to watch that the set F is a resolving set, it is adequate to confirm that ζ( |F) = ζ( |F) for any couple of distinguishable vertices , ∈ V(Φ) \ F.

The idea of resolving or locating set and that of metric dimension goes back to the 1950s. They were characterized by L. M. Blumenthal [START_REF] Blumenthal | Theory and applications of distance geometry[END_REF] with regards to metric space. These thoughts were acquainted with graph networks autonomously by Melter and Harary in 1976 [START_REF] Harary | On the metric dimension of a graph[END_REF], and P. J. Slater in 1975 [START_REF] Slater | Leaves of trees[END_REF]. Chart hypothesis has applications in numerous zones of figuring, social, and normal sciences and is likewise an affable play area for the investigation of the verification procedure in discrete science. Utilizations of this invariant to issues of picture preparing (or image processing) and design acknowledgment (or pattern recognition) are talked about in [START_REF] Melter | Metric bases in digital geometry[END_REF], to the route of exploring specialist (navigating agent or robots) in systems (or networks) are examined in [START_REF] Khuller | Localization in graphs[END_REF], applications to science are given in [START_REF] Chartrand | Resolvability in graphs and the metric dimension of a graph[END_REF], application to combinatorial enhancement (or optimization) is yielded in [START_REF] Sebo | On metric generators of graphs[END_REF], and to issues of check and system revelation (or network discovery) in [START_REF] Beerliova | Network discovery and verification[END_REF].

For an undirected graph Φ, the line graph of the graph Φ is a graph L(Φ) with vertex set V(L(Φ)) = E(Φ) and two different nodes f i and f j are adjacent in L(Φ) iff they have a common end vertex in Φ. Sometimes a line graph is also termed as edge graph, derived graph, or interchange graph. A polytope in elementary geometry is a geometric object with flat sides. When polytopes are having an additional property that they are convex sets and are contained in the n-dimensional space R n (Euclidean space), then they are termed as convex polytopes. Depending upon the problem with which we are dealing, polytopes and convex polytopes are defined accordingly (polytopes and convex polytopes may be defined in several ways by different researchers). Convex polytopes assume a significant job both in different branches of arithmetic and in applied zones, most quite in linear programming. The location number of a few classes of convex polytopes has been considered in ( [START_REF] Imran | Families of plane graphs with constant metric dimension[END_REF], [START_REF] Javaid | Families of regular graphs with constant metric dimension[END_REF], [START_REF] Sharma | Metric Dimension of heptagonal circular ladder[END_REF]). When every edge of the given undirected graph Φ is replaced by a path of length two, the graph obtained is known as the subdivision graph of the graph Φ, denoted by S(Φ).

Let ℵ establish a family of connected graphs (or charts). We call that the family ℵ has steady (or constant) location number if β(Ψ) = dim(Ψ) is free of the decision of the graph Ψ in ℵ and is finite (or limited). In other words, if all the graphs in ℵ have an indistinguishable location number, at that point ℵ is known as a family with a steady location number [START_REF] Tomescu | Metric dimension and R-Sets of a connected graph[END_REF]. In [START_REF] Chartrand | Resolvability in graphs and the metric dimension of a graph[END_REF] Chartrand et al. demonstrated that graphs (or charts) on n nodes have location number one iff it is a path ℘ n . Additionally, cycle C n has location number two for each positive integer n; n ≥ 3. Henceforth, C n (n ≥ 3) and ℘ n (n ≥ 2) establish a family of graphs (or charts) with a steady location number. Additionally, Petersen graphs P (n, 2) (generalized) and Harary graphs H 4,n , are also the families of graphs (or charts) with steady location numbers [START_REF] Javaid | Families of regular graphs with constant metric dimension[END_REF].

The join of two charts (or graphs)

Ψ 1 = Ψ 1 (V 1 , E 1 ) and Ψ 2 = Ψ 2 (V 2 , E 2 ) is a chart Φ = Φ(V, E) with the end goal that V = V 1 ∪ V 2 and E = E 1 ∪ E 2 ∪ { ς : ∈ V 1 and ς ∈ V 2 }. Then a fan F m is characterized as F m = K 1 + ℘ m for m ≥ 1, a wheel W m is characterized as W m = K 1 + C m ,
for m ≥ 3, and Jahangir graph J 2m (m ≥ 2) is gotten from the wheel graph W 2m by alternately deleting m spokes of the wheel graph (which is otherwise called as a gear graph). In [START_REF] Caceres | On the metric dimension of some families of graphs[END_REF], Caceres et al. decided the location number of the fan graph F m (m ≥ 1) which is 2m+2 5 for m / ∈ {1, 2, 3, 6}. Tomescu and Javaid [START_REF] Tomescu | On the metric dimension of the Jahangir graph[END_REF] acquired the location number of the Jahangir graph J 2m (m ≥ 4) which is for m / ∈ {3, 6}. Note that the location number of these three families (viz., Fan graph, Wheel graph, and Jahangir graph) of the plane graphs rely on the number of vertices in the graphs and in this way does not comprise the classes of plane graphs with steady location number (or constant metric dimensions). Now, a property in regards to the metric dimension two of a connected graph Ψ = Ψ(V, E) was demonstrated by Khuller et al. in [START_REF] Khuller | Landmarks in graphs[END_REF] and is Theorem 1. [START_REF] Khuller | Landmarks in graphs[END_REF] Let A ⊆ V(Ψ) be the basis set of the connected graph Ψ = Ψ(V, E) of cardinality two i.e., |A| = dim(Ψ) = β(Ψ) = 2, and say A = { , ξ}. Then, the following listed three points are true:

1. Between the vertices and ξ, there exists a unique and shortest path ℘.

2. The valencies (or degrees) of the nodes and ξ can never exceed 3.

3. The valency of any other node on ℘ can never exceed 5.

The main motivation in characterizing the classes of plane graphs with constant metric dimension (or with non-constant metric dimension) is that none of the plane graphs should be left with an unknown metric dimension (or location number). The radially symmetric plane graph ∆ n is obtained from the graph Γ n by the insertion of new edges in the graph Γ n between the nodes r t and q t+1 (1 t n), and is a graph on 6n edges and 4n vertices. We observed that V(Γ n ) = V(∆ n ), and in this study, we demonstrate that the location number of these two plane graphs (viz., Γ n and ∆ n ) are the same (i.e., β(Γ n ) = β(∆ n )) and constant. We also give the answers to the open problems that were raised in [START_REF] Sharma | Metric Dimension of heptagonal circular ladder[END_REF], by showing that there exists a minimal independent locating set with cardinality three for the plane graphs Γ n and Σ n . In what follows all lists t that do not fulfill the given disparities will be taken modulo n.

In the accompanying section, we acquire the location number of the radially symmetric plane graph ∆ n (see figure 1), which is obtained from the graph Γ n by the insertion of new edges between the nodes r t and q t+1 (1 t n), and for each positive integer n: n ≥ 6 we demonstrate that β(∆ n ) = 3. We also prove that the minimal resolving set obtained for the graph ∆ n is independent.

The Plane Graph ∆ n

The radially symmetric plane graph ∆ n is obtained from the graph Γ n by the insertion of new edges in the graph Γ n between the nodes r t and q t+1 (1 t n, see figure 1). At that point, the radially symmetrical plane graph ∆ n comprises of n 5 and 4-sided faces, a 2n-sided face, and an n-sided face and it consists of 6n edges and 4n number of nodes. By E(∆ n ) and V(∆ n ), we signify the arrangement of edges and vertices of a radially symmetrical plane graph ∆ n separately. Consequently, we have

V(∆ n ) = {p t , q t ,
r t , s t : 1 t n} and E(∆ n ) = {p t q t , q t r t , r t s t , p t p t+1 , s t r t+1 , r t q t+1 : 1 t n} For our motive, we call the cycle brought forth by the arrangement of vertices {p t : 1 t n} as the p-cycle, the arrangement of vertices {q t : 1 t n} as the set of core vertices, and the cycle brought forth by the arrangement of vertices {r t : 1 t n} ∪ {s t : 1 t n} as the rs-cycle. In the accompanying theorem, we discover that the location number of a radially symmetrical graph, ∆ n is 3 i.e., just 3 vertices properly chosen are adequate to determine all the vertices of the radially symmetrical graph ∆ n . Note that the choice of appropriate basis node is the crux of the problem. Theorem 2. Let n be a positive integer such that n 6 and ∆ n be the planar graph on 4n vertices as defined above. Then, we have dim(∆ n ) = 3 i.e., it has location number 3.

Proof. To demonstrate this, we eagerly consider the resulting two cases relying on the positive integer n i.e., when the positive whole number n is even and when it is odd. Case(I) When the integer n is even. For this situation, the integer n can be written as n = 2ϑ, where ϑ ∈ N and ϑ ≥ 3. Let R = {p 2 , p ϑ+1 , p n } ⊂ V(∆ n ). Now, in order to unveil that R is a locating or resolving set for the radially symmetrical plane graph ∆ n , we consign the metric codes for each vertex of V(∆ n ) R regarding the set R.

Presently, the metric codes for the vertices of p-cycle {p t : 1 t n} are

ζ(p t |R) =      (1, ϑ, 1), t = 1; (t -2, ϑ -t + 1, t), 3 ≤ t ≤ ϑ (2ϑ -t + 2, t -ϑ -1, 2ϑ -t), ϑ + 2 ≤ t ≤ 2ϑ -1.
The metric codes for the set of core nodes {q t : 1 t n} are

ζ(q t |R) =          (2, ϑ + 1, 2), t = 1; (t -1, ϑ -t + 2, t + 1), 2 ≤ t ≤ ϑ; (ϑ, 1, ϑ), t = ϑ + 1; (2ϑ -t + 3, t -ϑ, 2ϑ -t + 1), ϑ + 2 ≤ t ≤ 2ϑ.
At last, the metric codes for the nodes of rs-cycle {r t : 1 t n} ∪ {s t : 1 t n} are

ζ(r t |R) =                    (2, ϑ + 1, 3), t = 1; (t, ϑ -t + 2, t + 2), 2 ≤ t ≤ ϑ -1; (ϑ, 2, ϑ + 1), t = ϑ; (ϑ + 1, 2, ϑ), t = ϑ + 1; (2ϑ -t + 3, t -ϑ + 1, 2ϑ -t + 1), ϑ + 2 ≤ t ≤ 2ϑ -1; (2ϑ -t + 3, t -ϑ + 1, 2), t = 2ϑ. and ζ(s t |R) =                (3, ϑ + 1, 4), t = 1; (t + 1, ϑ -t + 2, t + 3), 2 ≤ t ≤ ϑ -1; (ϑ + 1, 3, ϑ + 1), t = ϑ; (2ϑ -t + 3, t -ϑ + 2, 2ϑ -t + 1), ϑ + 1 ≤ t ≤ 2ϑ -2; (2ϑ -t + 3, t -ϑ + 2, 3), 2ϑ -1 ≤ t ≤ 2ϑ.
We notice that no two vertices are having indistinguishable metric codes, suggesting that β(∆ n ) ≤ 3. Now, so as to finish the evidence for this case, we show that β(∆ n ) ≥ 3 by working out that there does not exist a resolving set R with the end goal that |R| = 2. Despite what might be expected, we guess that β(∆ n ) = 2. Now, by A 1 , A 2 , A 3 , and A 4 , we signify the arrangement of vertices as A 1 = {p t : 1 t n}, A 2 = {q t : 1 t n}, A 3 = {r t : 1 t n}, and A 4 = {s t : 1 t n}. At that point by Theorem 1, we find that the valency of basis nodes can never exceed 3. But except the vertices of the set A 3 , all other nodes of the radially symmetrical plane graph ∆ n have a valency less than or equals to 3. At that point, we have the accompanying prospects to be talked about.

When one, as well as the other node, are in the set A l ; l = 1, 2, 4.

Resolving sets

Contradictions {p 1 , p g }, p g (2 ≤ g ≤ n) For 2 ≤ g ≤ u, we have ζ(q 1 |{p 1 , p g }) = ζ(p n |{p 1 , p g }),
and when g = u + 1, we have

ζ(p 2 |{p 1 , p u+1 }) = ζ(p n |{p 1 , p u+1 }), a contradiction. {q 1 , q g }, q g (2 ≤ g ≤ n) For g = 2, we have ζ(s 1 |{q 1 , q 2 }) = ζ(s n |{q 1 , q 2 }), when g = 3, we have ζ(p 2 |{q 1 , q 3 }) = ζ(q 2 |{q 1 , q 3 }), when g = 4, we have ζ(s 2 |{q 1 , q 4 }) = ζ(r 3 |{q 1 , q 4 }), when 5 ≤ g ≤ u, we have ζ(s 1 |{q 1 , q g }) = ζ(p n |{q 1 , q g }),
and when g = u + 1, we have

ζ(p 2 |{q 1 , q u+1 }) = ζ(p n |{q 1 , q u+1 }), a contradiction. Resolving sets Contradictions {s 1 , s g }, s g (2 ≤ g ≤ n) For 2 ≤ g ≤ 3, we have ζ(s n |{s 1 , s g }) = ζ(q 1 |{s 1 , s g }), when 4 ≤ g ≤ 5, we have ζ(s 3 |{s 1 , s g }) = ζ(q 4 |{s 1 , s g }),
and when 6

≤ g ≤ u + 1, we have ζ(s 2 |{s 1 , s g }) = ζ(q 2 |{s 1 , s g }), a contradiction.
When one node is in the set A 1 and other lies in the set A l ; l = 2 and 4.

Resolving sets

Contradictions {p 1 , q g }, q g (1

≤ g ≤ n) For g = 1, u + 1, we have ζ(p n |{p 1 , q g }) = ζ(p 2 |{p 1 , q g }), when g = 2, we have ζ(s 2 |{p 1 , q 2 }) = ζ(s 1 |{p 1 , q 2 }),
and when 3

≤ g ≤ u + 1, we have ζ(s 2 |{p 1 , q g }) = ζ(q 2 |{p 1 , q g }), a contradiction. {p 1 , s g }, s g (1 ≤ g ≤ n) For g = 1, we have ζ(s n |{p 1 , s 1 }) = ζ(q 3 |{p 1 , s 1 }), when g = 2, we have ζ(q 3 |{p 1 , s 2 }) = ζ(s 1 |{p 1 , s 2 }), and when 3 ≤ g ≤ u -1, we have ζ(p n |{p 1 , s g }) = ζ(q 1 |{p 1 , s g }), when g = u, we have ζ(p n |{p 1 , s u }) = ζ(p 2 |{p 1 , s u }),
and when g = u + 1, we have

ζ(p 3 |{p 1 , s u+1 }) = ζ(q n |{p 1 , s u+1 }), a contradiction.
When one node is in the set A 2 and other lies in the set A 4 .

Resolving sets

Contradictions {q 1 , s g }, s g (1 ≤ g ≤ n)

For g = 1, we have

ζ(p n |{q 1 , s 1 }) = ζ(q 2 |{q 1 , s 1 }), when g = 2, we have ζ(q 2 |{q 1 , s 2 }) = ζ(s 1 |{q 1 , s 2 }), when 3 ≤ g ≤ 4 , we have ζ(s 2 |{q 1 , s g }) = ζ(q 3 |{q 1 , s g }), when 5 ≤ g ≤ u -1, we have ζ(s 1 |{q 1 , s g }) = ζ(p n |{q 1 , s g }), when g = u, we have ζ(p n |{q 1 , s u }) = ζ(p 2 |{q 1 , s u }),
and when g = u + 1, we have

ζ(p 3 |{q 1 , s u+1 }) = ζ(r n-1 |{q 1 , s u+1 }), a contradiction.
In this manner, the above conversation explains that there does not exist a resolving set comprising of two vertices for V(∆ n ) inferring that β(∆ n ) = 3 in this case.

Case(II)

When the integer n is odd. For this situation, the integer n can be written as n = 2ϑ + 1, where ϑ ∈ N and ϑ ≥ 3. Let R = {p 2 , p ϑ+1 , p n } ⊂ V(∆ n ). Now, in order to unveil that R is a locating or resolving set for the radially symmetrical plane graph ∆ n , we consign the metric codes for each vertex of V(∆ n ) R regarding the set R.

Presently, the metric codes for the vertices of p-cycle {p t : 1 t n} are

ζ(p t |R) =          (1, ϑ, 1), t = 1; (t -2, ϑ -t + 1, t), 3 ≤ t ≤ ϑ (ϑ, 1, ϑ -1), t = ϑ + 2; (2ϑ -t + 3, t -ϑ -1, 2ϑ -t + 1), ϑ + 3 ≤ t ≤ 2ϑ.
The metric codes for the set of core nodes {q t : 1 t n} are

ζ(q t |R) =                (2, ϑ + 1, 2), t = 1; (t -1, ϑ -t + 2, t + 1), 3 ≤ t ≤ ϑ (ϑ, 1, ϑ + 1), t = ϑ + 1; (ϑ + 1, 2, ϑ), t = ϑ + 2; (2ϑ -t + 4, t -ϑ, 2ϑ -t + 2), ϑ + 3 ≤ t ≤ 2ϑ + 1.
At last, the metric codes for the nodes of rs-cycle {r t : 1 t n} ∪ {s t : 1 t n} are

ζ(r t |R) =                (2, ϑ + 1, 3), t = 1; (t, ϑ -t + 2, t + 2), 2 ≤ t ≤ ϑ; (ϑ + 1, 2, ϑ + 1), t = ϑ + 1; (2ϑ -t + 4, t -ϑ + 1, 2ϑ -t + 2), ϑ + 2 ≤ t ≤ 2ϑ; (2ϑ -t + 4, t -ϑ + 1, 2), t = 2ϑ + 1. and ζ(s t |R) =                        (3, ϑ + 1, 4), t = 1; (t + 1, ϑ -t + 2, t + 3), 2 ≤ t ≤ ϑ -1; (ϑ + 1, 3, ϑ + 2), t = ϑ; (ϑ + 2, 3, ϑ + 1), t = ϑ + 1; (2ϑ -t + 4, t -ϑ + 2, 2ϑ -t + 2), ϑ + 2 ≤ t ≤ 2ϑ -1; (2ϑ -t + 4, t -ϑ + 2, 3), t = 2ϑ; (2ϑ -t + 4, ϑ + 2, 3), t = 2ϑ + 1.
Again we see that no two vertices are having indistinguishable metric codes, suggesting that β(∆ n ) ≤ 3. Now, on expecting that β(∆ n ) = 2, we consider that to be are parallel prospects as talked about in Case(I) and logical inconsistency can be inferred correspondingly. Consequently, β(∆ n ) = 3 for this situation too, which concludes the theorem.

This result can also be written as: Theorem 3. Let n be the positive integer such that n 6 and ∆ n be the planar graph on 4n vertices as defined above. Then, its independent location number is 3.

Proof. For proof, refer theorem 2.

In the next two sections, we prove that the minimal resolving sets obtained for the graphs Γ n and Σ n in [START_REF] Sharma | Metric Dimension of heptagonal circular ladder[END_REF] are independent and are of the same cardinality three i.e., |R| = 3 = β(Γ n ) = β(Σ n ).

The Plane Graph Γ n

The radially symmetric plane graph Γ n was defined in [START_REF] Sharma | Metric Dimension of heptagonal circular ladder[END_REF], and by E(Γ n ) and V(Γ n ), we signify the arrangement of edges and vertices of the graph Γ n separately. Consequently, we have V(Γ n ) = {p t , r t , q t , s t : 1 t n} and E(Γ n ) = {p t q t , q t r t , r t s t , p t p t+1 , s t r t+1 : 1 t n} For our motive, we call the cycle brought forth by the arrangement of vertices {p t : 1 t n} as the p-cycle, the arrangement of vertices {q t : 1 t n} as the set of core vertices, and the cycle brought forth by the arrangement of vertices {r t : 1 t n}∪{s t : 1 t n} as the rs-cycle. In the accompanying theorem, we discover that the independent location number of a radially symmetrical graph, Γ n is 3 i.e., just 3 vertices properly chosen are adequate to determine all the vertices of the radially symmetrical graph Γ n . Note that the choice of appropriate basis node is the crux of the problem.

Theorem 4. Let n be the positive integer such that n 6 and Γ n be the planar graph on 4n vertices as defined above. Then, its independent location number is 3.

Proof. To demonstrate this, we eagerly consider the resulting two cases relying on the positive integer n i.e., when the positive whole number n is even and when it is odd. Case(I) When the integer n is even. For this situation, the integer n can be written as n = 2ϑ, where ϑ ∈ N and ϑ ≥ 3. Let R = {p 2 , p ϑ+1 , p n } ⊂ V(Γ n ). Now, in order to unveil that R is a locating or resolving set for the radially symmetrical graph Γ n , we consign the metric codes for each vertex of V(Γ n ) R regarding the set R.

Presently, the metric codes for the vertices of p-cycle {p t : 1 t n} are

ζ(p t |R) =      (1, ϑ, 1), t = 1; (t -2, ϑ -t + 1, t), 3 ≤ t ≤ ϑ (2ϑ -t + 2, t -ϑ -1, 2ϑ -t), ϑ + 2 ≤ t ≤ 2ϑ -1.
The metric codes for the set of core nodes {q t : 1 t n} are

ζ(q t |R) =          (2, ϑ + 1, 2), t = 1; (t -1, ϑ -t + 2, t + 1), 2 ≤ t ≤ ϑ; (ϑ, 1, ϑ), t = ϑ + 1; (2ϑ -t + 3, t -ϑ, 2ϑ -t + 1), ϑ + 2 ≤ t ≤ 2ϑ.
At last, the metric codes for the nodes of rs-cycle {r t : 1 t n} ∪ {s t : 1 t n} are

ζ(r t |R) =          (3, ϑ + 2, 3), t = 1; (t, ϑ -t + 3, t + 2), 2 ≤ t ≤ ϑ; (ϑ + 1, 2, ϑ + 1), t = ϑ + 1; (2ϑ -t + 4, t -ϑ + 1, 2ϑ -t + 2), ϑ + 2 ≤ t ≤ 2ϑ. and ζ(s t |R) =                    (3, ϑ + 2, 4), t = 1; (t + 1, ϑ -t + 3, t + 3), 2 ≤ t ≤ ϑ -1; (ϑ + 1, 3, ϑ + 2), t = ϑ; (ϑ + 2, 3, ϑ + 1), t = ϑ + 1; (2ϑ -t + 4, t -ϑ + 2, 2ϑ -t + 2), ϑ + 1 ≤ t ≤ 2ϑ -1; (2ϑ -t + 4, t -ϑ + 2, 3), t = 2ϑ.
We notice that no two vertices are having indistinguishable metric codes, suggesting that β(Γ n ) ≤ 3. Now, so as to finish the evidence for this case, we show that β(Γ n ) ≥ 3 by working out that there does not exist a locating or resolving set R with the end goal that |R| = 2. Despite what might be expected, we guess that β(Γ n ) = 2. At that point, there are indistinguishable prospects from talked about in the Case(I) of Theorem 2 in [START_REF] Sharma | Metric Dimension of heptagonal circular ladder[END_REF] and inconsistency can be concluded likewise.

In this manner, the above conversation explains that there does not exist a resolving set comprising of two vertices for V(Γ n ) inferring that β(Γ n ) = 3 in this case.

Case(II)

When the integer n is odd. In this case, the integer n can be written as n = 2ϑ + 1, where ϑ ∈ N and ϑ ≥ 3. Let R = {p 2 , p ϑ+1 , p n } ⊂ V(Γ n ). Now, in order to unveil that R is a locating (or resolving) set for the radially symmetrical plane graph Γ n , we consign the metric codes for each vertex of V(Γ n ) R regarding the set R.

Presently, the metric codes for the vertices of p-cycle {p t : 1 t n} are

ζ(p t |R) =          (1, ϑ, 1), t = 1; (t -2, ϑ -t + 1, t), 3 ≤ t ≤ ϑ (ϑ, 1, ϑ -1), t = ϑ + 2; (2ϑ -t + 3, t -ϑ -1, 2ϑ -t + 1), ϑ + 3 ≤ t ≤ 2ϑ.
The metric codes for the set of core nodes {q t : 1 t n} are

ζ(q t |R) =          (2, ϑ + 1, 2), t = 1; (t -1, ϑ -t + 2, t + 1), 3 ≤ t ≤ ϑ (ϑ + 1, 2, ϑ), t = ϑ + 2; (2ϑ -t + 4, t -ϑ, 2ϑ -t + 2), ϑ + 3 ≤ t ≤ 2ϑ + 1.
At last, the metric codes for the nodes of rs-cycle {r t : 1 t n} ∪ {s t : 1 t n} are

ζ(r t |R) =          (3, ϑ + 2, 3), t = 1; (t, ϑ -t + 3, t + 2), 3 ≤ t ≤ ϑ (ϑ + 2, 3, ϑ + 1), t = ϑ + 2; (2ϑ -t + 5, t -ϑ + 1, 2ϑ -t + 3), ϑ + 3 ≤ t ≤ 2ϑ + 1. and ζ(s t |R) =                (3, ϑ + 2, 4), t = 1; (t + 1, ϑ -t + 3, t + 3), 2 ≤ t ≤ ϑ; (ϑ + 2, 3, ϑ + 2), t = ϑ + 1; (2ϑ -t + 5, t -ϑ + 2, 2ϑ -t + 3), ϑ + 2 ≤ t ≤ 2ϑ; (2ϑ -t + 5, t -ϑ + 2, 3), t = 2ϑ + 1.
Again we see that no two vertices are having indistinguishable metric codes, suggesting that β(Γ n ) ≤ 3. Now, on expecting that β(Γ n ) = 2, we consider that to be are parallel prospects as talked about in Case(I) and logical inconsistency can be inferred correspondingly. Consequently, β(Γ n ) = 3 for this situation too, which concludes the theorem.

The Plane Graph Σ n

The radially symmetric plane graph Σ n was defined in [START_REF] Sharma | Metric Dimension of heptagonal circular ladder[END_REF], and by E(Σ n ) and V(Σ n ), we signify the arrangement of edges and vertices of the graph Σ n separately. Consequently, we have V(Σ n ) = {p t , r t , q t , s t : 1 t n} and E(Σ n ) = {p t q t , q t r t , r t s t , p t p t+1 , s t r t+1 , r t r t+1 : 1 t n} For our motive, we call the cycle brought forth by the arrangement of vertices {p t : 1 t n} as the p-cycle, the arrangement of vertices {q t : 1 t n} as the set of core vertices, and the cycle brought forth by the arrangement of vertices {r t : 1 t n} ∪ {s t : 1 t n} as the rs-cycle. In the accompanying Theorem, we discover that the independent location number of a radially symmetrical graph, Σ n is 3 i.e., just 3 vertices properly chosen are adequate to determine all the vertices of the radially symmetrical graph Σ n . Note that the choice of appropriate basis node is the crux of the problem.

ζ(p t |R) =          (1, ϑ, 1), t = 1; (t -2, ϑ -t + 1, t), 3 ≤ t ≤ ϑ (ϑ, 1, ϑ -1), t = ϑ + 2; (2ϑ -t + 3, t -ϑ -1, 2ϑ -t + 1), ϑ + 3 ≤ t ≤ 2ϑ.
The metric codes for the set of core nodes {q t : 1 t n} are Again we see that no two vertices are having indistinguishable metric codes, suggesting that β(Σ n ) ≤ 3. Now, on expecting that β(Σ n ) = 2, we consider that to be are parallel prospects as talked about in Case(I) and logical inconsistency can be inferred correspondingly. Consequently, β(Σ n ) = 3 for this situation too, which concludes the theorem.

ζ(q t |R) =          (2,

Conclusion

In this paper, we got another family of plane graph viz., ∆ n , which is acquired by the expansion of new edges in a radially symmetrical graph Γ n [START_REF] Sharma | Metric Dimension of heptagonal circular ladder[END_REF], and demonstrate that V(Γ n ) = V(∆ n ) and dim(Γ n ) = dim(∆ n ). We likewise offer responses to the problems brought up in [START_REF] Sharma | Metric Dimension of heptagonal circular ladder[END_REF], that is there exist a self-governing (or an independent) locating set for the two classes of the radially symmetrical graphs (viz., Γ n , and Σ n ). We have furthermore observed that the location number of these three classes of the radially symmetrical graphs viz., ∆ n , Γ n , and Σ n , is finite (or limited) and is independent (or free) of the number of vertices of these radially symmetrical graphs, and just 3 vertices sensibly picked are agreeable to choose all the vertices of these classes of radially symmetrical plane graphs. We besides saw that the basis set R is self-governing (or independent) for these three radially even families of plane graphs.

2m 3 ,

 3 and in [3] Chartrand et al. decided the location number of the wheel chart (or graph) W m (m ≥ 3) which is 2m+2 5

Figure 1 :

 1 Figure 1: The radially symmetric graph ∆ n

Figure 2 :

 2 Figure 2: The radially symmetric graph Γ n

Figure 3 :

 3 Figure 3: The radially symmetrical graph Σ n

  ϑ + 1, 2), t = 1; (t -1, ϑ -t + 2, t + 1), 3 ≤ t ≤ ϑ (ϑ + 1, 2, ϑ), t = ϑ + 2; (2ϑ -t + 4, t -ϑ, 2ϑ -t + 2), ϑ + 3 ≤ t ≤ 2ϑ + 1.At last, the metric codes for the nodes of rs-cycle {r t : 1 t n} ∪ {s t : 1 t n} areζ(r t |R) = -t + 3, t + 2), 3 ≤ t ≤ ϑ (ϑ + 2, 3, ϑ + 1), t = ϑ + 2; (2ϑ -t + 5, t -ϑ + 1, 2ϑ -t + 3), ϑ + 3 ≤ t ≤ 2ϑ + 1. , ϑ -t + 3, t + 3), 2 ≤ t ≤ ϑ; (ϑ + 2, 3, ϑ + 2), t = ϑ + 1; (2ϑ -t + 5, t -ϑ + 2, 2ϑ -t + 3), ϑ + 2 ≤ t ≤ 2ϑ; (2ϑ -t + 5, t -ϑ + 2, 3), t = 2ϑ + 1.

	         (t, ϑ and (3, ϑ + 2, 3),	t = 1;
	ζ(s t |R) =	  (3, ϑ + 2, 4),       (t + 1	t = 1;
		      

Theorem 5. Let n be a positive integer such that n 6 and Σ n be the planar graph on 4n vertices as defined above. Then, its independent location number is 3.

Proof. To demonstrate this, we consider the resulting two cases relying on the positive integer n i.e., when the positive whole number n is even and when it is odd. Case(I) When the integer n is even. For this Situation, the integer n can be written as n = 2ϑ, where ϑ ∈ N and ϑ ≥ 3. Let R = {p 2 , p ϑ+1 , p n } ⊂ V(Σ n ). Now, in order to unveil that R is a locating or resolving set for the radially symmetrical graph Σ n , we consign the metric codes for each vertex of V(Σ n ) R regarding the set R.

presently, the metric codes for the vertices of p-cycle {p t : 1 t n} are

The metric codes for the set of core nodes {q t : 1 t n} are

At last, the metric codes for the nodes of rs-cycle {r t : 1 t n} ∪ {s t : 1 t n} are

We notice that no two vertices are having indistinguishable metric codes, suggesting that β(Σ n ) ≤ 3. Now, so as to finish the evidence for this case, we show that β(Σ n ) ≥ 3 by working out that there does not exist a locating or resolving set R with the end goal that |R| = 2. Despite what might be expected, we guess that β(Σ n ) = 2. At that point, there are indistinguishable prospects from talked about in the Case(I) of Theorem 3 in [START_REF] Sharma | Metric Dimension of heptagonal circular ladder[END_REF] and inconsistency can be concluded likewise.

In this manner, the above conversation explains that there does not exist a resolving set comprising of two vertices for V(Σ n ) inferring that β(Σ n ) = 3 in this case.

Case(II)

When the integer n is odd. For this situation, the integer n can be written as n = 2ϑ + 1, where ϑ ∈ N and ϑ ≥ 3. Let R = {p 2 , p ϑ+1 , p n } ⊂ V(Σ n ). Now, to unveil that R is a locating or resolving set for the radially symmetrical graph Σ n , we consign the metric codes for each vertex of V(Σ n ) R regarding the set R.

Presently, the metric codes for the vertices of p-cycle {p t : 1 t n} are


PhD Sarah Guimbal

MS Lauriane Cornuault

MS Paul Rouault

PhD Pierre-Louis Hollier

PharmD Candice Chapouly

PharmD Marie-Lise Bats

MD Julien Imbault

PhD Alain- Pierre Gadeau

MD, PhD Thierry Couffinhal

PhD Marie-Ange Renault
email: marie-ange.renault@inserm.fr

Mast cells are the trigger of small vessel disease and diastolic dysfunction in diabetic obese mice

Keywords: Mast cells, endothelial dysfunction, diastolic dysfunction, cardiovascular risk factors Animal Models of Human Disease, Diabetes, Type 2, Heart Failure, Coronary Circulation, Endothelium

Objective: Heart failure with preserved ejection fraction (HFpEF) is a major healthcare issue which has been difficult to manage to date, due to its complex and not well understood pathophysiology. Specifically, if a wealth of literature focuses on HFpEF cardiac component, very little information can be found on its vascular component. Our goal is to unravel the critical role of cardiac small vessel disease in the pathophysiology of diastolic dysfunction.

Approach and Results:

To do so, we used Leptin receptor deficient (Lepr db/db ) female mice, a recognized model of diastolic dysfunction. In these mice, the increased end diastolic pressure (EDP) signing diastolic dysfunction is associated with vascular leakage, endothelial cell activation and leucocyte infiltration. Strikingly, a RNA sequencing analysis of the cardiac vascular fraction of both Lepr db/db and control mice confirmed endothelial dysfunction and systemic inflammation, but above all, revealed a strong increase in several mast cell markers (notably FceR1a, Tryptase and Chymase). We then confirmed this accumulation of activated mast cells in the heart of Lepr db/db mice via histology. Importantly, we found that both mast cell degranulation inhibition and specific histamine blockade reduced vascular leakage, leucocyte infiltration and subsequently EDP in Lepr db/db mice. This demonstrated, for the first time, that mast cells, via histamine release, participate in the development of cardiac small vessel disease leading to diastolic dysfunction. Conclusions: Cardiac small vessel disease is a key mechanism of HFpEF pathophysiology that can be targeted to prevent the occurrence of diastolic dysfunction, by using both mast cell stabilizers and antihistaminic drugs.

Fcer1a

High affinity immunoglobulin epsilon receptor subunit alpha H1R Histamine H1 receptor SMA Alpha Smooth Muscle Actin NO nitric oxide

Introduction

A significant proportion of patients with clinical syndrome of heart failure (HF) happens to have a normal ventricular ejection fraction (EF) referred to as heart failure with preserved ejection fraction (HFpEF) as opposed to patients with reduced ejection fraction (HFrEF) [START_REF] Pfeffer | Heart Failure With Preserved Ejection Fraction In Perspective[END_REF] .

HFpEF currently accounts for about 50% of all HF patients, and its prevalence relative to

HFrEF is rising at a rate of approximately 1% per year. HFpEF patients are generally older, more often female and have a high prevalence of cardiovascular and non-cardiovascular comorbidities, such as obesity, metabolic syndrome, diabetes mellitus type 2, hypertension, atrial fibrillation or renal dysfunction [START_REF] Pfeffer | Heart Failure With Preserved Ejection Fraction In Perspective[END_REF] . HFpEF is diagnosed in the presence of HF signs and/or symptoms, preserved systolic left ventricular (LV) function, with an LV ejection fraction >50% and LV end-diastolic volume index <97 ml/m² with evidence of diastolic LV dysfunction [START_REF] Sharma | Heart failure with preserved ejection fraction: mechanisms, clinical features, and therapies[END_REF] . In recent years, there have been significant advances in understanding aetiologies and mechanisms of HFrEF, leading to new therapeutic options. In contrast, if some progresses on HFpEF pathophysiology have been made, therapeutic targets are still missing and no therapies are available to reduce hospitalization or mortality for HFpEF patients. There is, therefore, an urgent need for a better understanding of HFpEF pathophysiology in order to develop new drugs [START_REF] Lyle | HFpEF, a Disease of the Vasculature: A Closer Look at the Other Half[END_REF] . A unifying, however untested, theory of the pathophysiology of HFpEF suggests that comorbidities lead to a systemic low grade inflammation, which triggers coronary microvascular dysfunction 2,4-6 and subsequently heart failure.

Endothelial dysfunction mostly refers to impaired endothelium-dependent, nitric oxidemediated vasorelaxation. Nevertheless, endothelial dysfunction actually regroups an ensemble of features that includes acquirement of a pro-inflammatory /pro-thrombotic phenotype due to an increased expression of adhesion and pro-thrombotic molecules such as vascular cell adhesion molecule-1 (Vcam1) and intercellular adhesion molecule-1 (Icam1), abnormal vascular leakage due to altered endothelial intercellular junctions and endothelial dedifferentiation promoting endothelial to mesenchymal transition [START_REF] Li | Reassessing endothelial-to-mesenchymal transition in cardiovascular diseases[END_REF] . EC dysfunction may then lead to a compromised heart perfusion because of impaired nitric oxide (NO)dependent vasodilatation, cardiac oedema or thrombi formation. Besides, it may also promote cardiac fibrosis and inflammation [START_REF] Marti | Endothelial dysfunction, arterial stiffness, and heart failure[END_REF][START_REF] Marechaux | Vascular and Microvascular Endothelial Function in Heart Failure With Preserved Ejection Fraction[END_REF][START_REF] D'amario | Microvascular Dysfunction in Heart Failure With Preserved Ejection Fraction[END_REF] . Finally, an impaired production of endothelial-derived active substances as NO, endothelin-1, angiotensin II, Neuregulin is also proposed to directly affect cardiomyocyte passive tension [START_REF] Tirziu | Cell communications in the heart[END_REF][START_REF] Noireaud | Recent insights in the paracrine modulation of cardiomyocyte contractility by cardiac endothelial cells[END_REF][START_REF] Schiattarella | Nitrosative stress drives heart failure with preserved ejection fraction[END_REF] .

What is known so far is that HFpEF is associated with an increased endothelial expression of ICAM-1 and E-selectin, an increased oxidative stress and a decreased nitric oxide synthase 3 (NOS3) activity both in animal models and in patients [START_REF] Franssen | Myocardial Microvascular Inflammatory Endothelial Activation in Heart Failure With Preserved Ejection Fraction[END_REF] . Also, brachial artery flow-mediated dilation assessment and laser Doppler flow measurement have revealed an impaired vascular and microvascular function in patients with HFpEF [START_REF] Marechaux | Vascular and Microvascular Endothelial Function in Heart Failure With Preserved Ejection Fraction[END_REF] . Finally, HFpEF is typically associated with increased myocardial stiffening (including increased cardiomyocyte passive stiffness and accumulation of stiff and resistant collagen fibres). Nevertheless the causal role of endothelial dysfunction in the pathophysiology of HFpEF has never been established.

In the present study we used Leptin receptor (Lepr) db/db female mice as a model of HFpEF to further characterize the phenotype and mechanisms of cardiac microvessel dysfunction associated with diastolic dysfunction. Notably, Lepr db/db female mice recapitulate the main risk factors for HFpEF i.e. obesity, diabetes and female sex. As a major breakthrough, this study demonstrates for the first time that abnormally increased mast cell activation participates in the pathophysiology of both cardiac microvessel disease and diastolic dysfunction in Lepr db/db female mice.

Methods

The authors declare that all supporting data are available within the article and its online supplementary files.

Mice

Lepr db mice (BKS.Cg-Dock7 m /+ Lepr db /+J) were obtained from Charles River laboratories and bred together in our animal facility to obtain Lepr db/db and control Lepr db /+ mice.

Animal experiments were performed in accordance with the guidelines from Directive 2010/63/EU of the European Parliament on the protection of animals used for scientific purposes and approved by the local Animal Care and Use Committee of Bordeaux University.

Only females were used. Mice were either sacrificed by cervical dislocation or exsanguination under deep anesthesia (ketamine 100 mg/kg and xylazine 20 mg/kg, IP).

Cromolyn Sodium/Cetirizine therapy

To prevent mast cell degranulation, mice were treated with 50 mg/kg/day cromolyn sodium (Abcam) via intra-peritoneal injections for 28 days. Untreated mice received 0,9% NaCl daily intra-peritoneal injections. To investigate the role of Histamine release by mast cell, mice were treated with 4 mg/kg/day cetirizine (Arrow Génériques) orally (in the drinking water) for 28 days.

Echocardiography

Left-ventricular ejection fraction and LV dimension were measured on a high-resolution echocardiographic system equipped with a 30-MHz mechanical transducer (VEVO 2100, VisualSonics Inc.) as previously described [START_REF] Renault | The Hedgehog transcription factor Gli3 modulates angiogenesis[END_REF][START_REF] Roncalli | Sonic hedgehog-induced functional recovery after myocardial infarction is enhanced by AMD3100-mediated progenitor-cell mobilization[END_REF] . Mice were anchored to a warming platform in a supine position, limbs were taped to the echocardiograph electrodes, and chests were shaved and cleaned with a chemical hair remover to minimize ultrasound attenuation.

UNI'GEL ECG (Asept Inmed), from which all air bubbles had been expelled, was applied to the thorax to optimize the visibility of the cardiac chambers. Ejection fractions were evaluated by planimetry as recommended [START_REF] Schiller | Recommendations for quantitation of the left ventricle by two-dimensional echocardiography. American Society of Echocardiography Committee on Standards, Subcommittee on Quantitation of Two-Dimensional Echocardiograms[END_REF] . Two-dimensional, parasternal long-axis and short-axis views were acquired, and the endocardial area of each frame was calculated by tracing the endocardial limits in the long-axis view, then the minimal and maximal areas were used to determine the left-ventricular end-systolic (ESV) and end-diastolic (EDV) volumes, respectively. The system software uses a formula based on a cylindricalhemiellipsoid model (volume=8.area²/3π/ length) [START_REF] Mohan | Cross sectional echocardiographic left ventricular ejection fraction: method based variability[END_REF] . The left-ventricular ejection fraction was derived from the following formula: (EDV-ESV)/EDV*100. The cardiac wall thickness, left ventricular posterior wall (LVPW), inter-ventricular septum (IVS) and left ventricular internal diameter (LVID) were calculated by tracing wall limits in both the long and short axis views.

LV pressure /systolic blood pressure measurement LV diastolic pressure measurement was assessed via invasive catheterization technique.

Briefly, mice were anesthetized with Isoflurane. A Scisense pressure catheter (Transonic) was inserted into the LV through the common carotid artery. Pressure was recorded using LabChart software. End diastolic pressure, dP/dt minimum and maximum, Tau and heart rate were automatically calculated by a curve fit through end-systolic and end-diastolic points on the pressure plot.

Lung edema assessment

Lung edema was quantified using the wet/dry weight ratio. The wet weight was measured immediately after the lung was harvested, while the dry weight was measured after the lung was incubated for 72 hours at 85°C.

Immuno-histological assessments

Prior to staining, heart were stopped in diastole using KCl, perfused and then fixed with 10% formalin for 4 hours, paraffin embedded and cut into 7 µm thick sections. Alternatively, heart were fresh frozen in OCT, then cut into 7 µm thick sections.

ECs were identified using rat anti-CD31 antibodies (Histonova, cat# DIA-310) or FITC-labelled Isolectin B4 (Sigma Aldrich, Cat#L2895). Smooth muscle cells were identified using Cy3conjugated mouse anti-Alpha Smooth Muscle Actin (SMA) antibodies (Sigma Aldrich, Cat#C6198). Albumin was stained using sheep anti-albumin antibodies (Abcam, Cat# ab8940). Pan-leucocytes were identified using rat anti-mouse CD45 antibodies (BD Pharmingen Inc, Cat# 550539). Macrophages were identified using rat anti-CD68 antibodies (Biolegend, Cat# 137001). M2 macrophages were identified using goat anti-mannose

Receptor C-Type 1 (Mrc1) antibodies (R&D systems, Cat# AF1556). Neutrophils were identified using rat anti-Ly6G antibodies (BD Pharmingen Inc, Cat# 551459). T lymphocytes were identified using goat anti-CD3 antibodies (Santa Cruz Biotechnology, Inc, Cat# sc-1127).

B-cells were stained using rat anti-B220/CD45R antibodies (R&D systems, Cat# MAB1217)

Mast/Stem Cell Growth Factor Receptor c-Kit was identified using goat anti-CD117 antibodies (R&D systems, Cat# AF1356). Fcer1a was identified using rabbit anti-Fcer1a antibodies (Biorbyt, Cat# orb41809). Cardiomyocytes were immunostained using rabbit antidesmin antibodies (DB 148, Cat#DB148), mouse anti-sarcomeric actin antibodies (Sigma, Cat# A2172) and rabbit anti-connexin-43 (Cx43) antibodies (Sigma, Cat# C6219). Endothelial adherens junction integrity was assessed using mouse anti-Cadherin-5 (Cdh5) antibodies (Santa Cruz Biotechnology, Inc, Cat# sc-9989).

For immunofluorescence analyzes, primary antibodies were resolved with Alexa Fluor®conjugated secondary polyclonal antibodies (Invitrogen, Cat# A-21206, A-21208, A-11077, A-11057, A-31573, A-10037 ) and nuclei were counterstained with DAPI (1/5000). Negative controls using secondary antibodies only were done to check for antibody specificity.

Cardiomyocyte membranes were stained using Wheat Germ Agglutinin (WGA), Alexa Fluor™ 488 Conjugate (Invitrogen).

Fibrosis was quantified after Sirius red staining of heart sections by quantifying the percentage of red-positive area. Cardiomyocyte mean surface area was measured using the Image J software after membrane staining with Wheat Germ Agglutinin (WGA), Alexa Fluor™ 488 Conjugate (Invitrogen) in 5 pictures randomly taken under x630 magnification. Pictures were taken in areas where cardiomyocytes were oriented transversally. Capillary density (CD31+ vessels) was quantified in 4 pictures taken under x260 magnification in areas where cardiomyocytes were oriented transversally.

To assess the mean capillary diameter, the diameter of 10 capillaries randomly chosen in each picture was measured via Image J. Inflammatory cell density (CD45+, CD68+, CD3+ and Ly6G+) was quantified in 8 pictures randomly taken under x260 magnification. Mast cell density and mast cell degranulation was assessed after Toluidine blue staining of heart sections. Mast cells were counted in the entire section.

All pictures and quantifications (done using ImageJ/Fiji v2.0.0-rc-59 software (National Institute of Health, USA)) were performed by a blinded investigator. More precisely, all samples were assigned a random number prior to animal sacrifice, data collection and analysis. At the end of the experiment, the genotype/treatment for each animal was unveiled to allow data comparison and experimental conclusion.

Measurement of Reactive Oxygen Species (ROS)

Fresh frozen heart sections were incubated with 1 µmol/L dihydroethidium (Sigma Aldrich, Cat# 37291) for 10 minutes at room temperature. Red fluorescence was quantified in 8 pictures randomly taken under x400 magnification using Image J software.

Quantitative RT-PCR

RNA was isolated using Tri Reagent® (Molecular Research Center Inc) as instructed by the manufacturer, from heart tissue that had been snap-frozen in liquid nitrogen and homogenized. For quantitative RT-PCR analyses, total RNA was reverse transcribed with M-MLV reverse transcriptase (Promega) and amplification was performed on an AriaMx Real Time PCR system (Agilent Technologies) using B-R SYBER® Green SuperMix (Quanta Biosciences). Primer sequences are reported in Supplementary table 1.

The relative expression of each mRNA was calculated by the comparative threshold cycle method and normalized to 18S rRNA expression.

Soluble/insoluble collagen measurement

Total cardiac collagen and insoluble cardiac collagen were quantified using the Sircol™ Soluble Collagen Assay and The Sircol™ INSOLUBLE Collagen Assay (Biocolor) according to the manufacturer's instructions.

Western blot analysis

Plamatic IgE level was evaluated by SDS PAGE using rat anti-mouse IgE antibodies (R&D systems Cat# MAB9935). Nos3 and phosphorylated Nos3 levels were evaluated by SDS PAGE using rabbit anti-Nos3 (Cell signaling technology, Cat#32027) and rabbit anti phospho-Nos3 (S1177) (Cell signaling technology, Cat#9571) antibodies respectively. Icam1, Vcam1, Chemokine (C-C motif) ligand 7 (Ccl7) and Interleukin-6 (Il-6) protein levels were evaluated by SDS PAGE using goat anti-Icam-1 (R&D systems Cat# AF-796), rabbit anti-Vcam-1 (abcam, Cat# ab134047), goat anti-Ccl7 (R&D systems Cat# AF-456) and goat anti-Il-6 (R&D systems Cat# AF-406) antibodies respectively. Protein loading quantity was controlled using mouse monoclonal anti-a-tubulin antibodies (Sigma, Cat# T5168).

Statistics

Results are reported as mean ± SEM. Comparisons between groups were analyzed for significance with the non-parametric Mann-Whitney test, the Kruskal-Wallis test followed by Dunn's multiple comparison test (for more than two groups) or a 2 way ANOVA followed by Sidak's multiple comparison test (for kinetics analyses) using GraphPad Prism v8.0.2 (GraphPad Inc, San Diego, Calif). The normality and variance were not tested. Differences between groups were considered significant when p≤0.05 (*: p≤0.05; **: p≤0.01; ***: p≤0.001).

Results

Lepr db/db female mice have diastolic dysfunction from 3 months of age.

First we quantified and characterized the kinetics of appearance of cardiovascular risk factors in the Lepr db/db female mice and in their control Lepr db /+ littermates bred in our laboratory. Lepr Then we assessed cardiac function in these mice mainly via echocardiography and LV catheterization. Lepr db/db female mice have normal ejection fraction (≥50%) during at least their first year of life (Figure 1A). However, they display diastolic dysfunction attested by a significantly increased end diastolic pressure (EDP) from 3 months of age (Figure 1B) and an increased lung water content signing pulmonary edema (Figure 1C). However, the maximum and minimum dP/dt, the relaxation time constant Tau and the heart rate were not modified at rest (Supplemental Figure II A-D). Increased EDP was associated with increased heart weight/tibia length ratio and increased diastolic LV posterior wall thickness attesting cardiac hypertrophy (Figure 1D-E). Diastolic LV internal diameter was not different from that of control mice (Figure 1F). Consistent with cardiac hypertrophy, cardiomyocyte were significantly larger in 3 month old Lepr db/db female mice compared to control mice (Figure 1G, H). Myocyte dedifferentiation, attested by myosin heavy chain 7 (Myh7) over expression appears later on at 6 months of age (Figure 1I). Notably, while both cardiac atrial natriuretic peptide (ANP) and brain natriuretic peptide (BNP) mRNA expression increased with age, they were not different in Lepr db/db mice and control mice (Supplemental Figure II E-F).

Altogether, this first set of data confirmed that Lepr db/db mice have diastolic dysfunction with preserved systolic function and may be used as a mouse model of HFpEF. Notably this is consistent with previous investigations [START_REF] Alex | Characterization of a mouse model of obesity-related fibrotic cardiomyopathy that recapitulates features of human heart failure with preserved ejection fraction[END_REF][START_REF] Horgan | Murine models of diastolic dysfunction and heart failure with preserved ejection fraction[END_REF] .

Lepr db/db female mice do not have cardiomyocyte gross anomalies.

To search for cardiomyocyte major anomalies, we first performed Desmin and sarcomeric alpha actin staining on heart cross sections to visualize cardiomyocyte sarcomeres and Cx43 staining to identify intercalary disks. We did not observe any significant difference between In conclusion, Lepr db/db female mice do not have cardiomyocyte gross anomalies, which is consistent with the unchanged dP/dT minimum and maximum.

Lepr db/db female mice do not have significant cardiac fibrosis.

Cardiac fibrosis, often associated with HFpEF, was first evaluated via picroSirius red staining.

As shown in supplemental figure III, the red staining was equivalent in 3 month old Lepr db/db In conclusion, diastolic dysfunction in Lepr db/db female mice is not associated with increased cardiac fibrosis.

3 month old Lepr db/db female mice have cardiac microvessel disease.

To phenotype the cardiac microvessel of 3 month old Lepr db/db female mice, we performed both immuno-histological and gene expression analyses. Gene expression analysis was performed by a transcriptomic analysis via RNA sequencing of the cardiac vascular fraction which includes ECs, pericytes, few smooth muscle cells and perivascular inflammatory cells (Supplemental Table II). The main RNA sequencing results were confirmed by RT-qPCR performed on total heart RNA samples.

First we quantified cardiac capillary density and found that capillary density is significantly decreased in the heart of Lepr db/db female mice (Figure 2A-B) which is fully consistent with previous analyses performed on human patients with HFpEF [START_REF] Mohammed | Coronary microvascular rarefaction and myocardial fibrosis in heart failure with preserved ejection fraction[END_REF] . Besides, we found that the mean capillary diameter was significantly increased (Figure 2C) (notably cardiac capillaries are not muscularized), while the diameter of SMA+ arterioles was slightly but not significantly decreased (Figure 2D-E Altogether, these results demonstrate that Lepr db/db have cardiac microvessel disease characterized by a decreased capillary density but an increased capillary diameter.

Moreover, they display a pro-inflammatory and a pro-coagulant phenotype.

3 month old Lepr db/db female mice have cardiac inflammation.

In association with upregulation of multiple inflammatory cytokines, leucocyte infiltration in the cardiac tissue of Ler db/db mice was significantly increased (Figure 3G-H). Among However, T-lymphocytes and neutrophils densities were not different between Lepr db/db mice and their control littermates.

month old Lepr db/db female mice have abnormal cardiac mast cell activation.

Apart from highlighting EC dysfunction and cardiac inflammation, transcriptomic analysis of the cardiac vascular fraction also revealed that among the 50 genes with log2FC≥2, 7 genes were mast cell markers (Figure 4A-B). We confirmed Carboxypeptidase A3, mast cell (Cpa3)

and High affinity immunoglobulin epsilon receptor subunit alpha (Fcer1a) overexpression in total heart extracts via RT-qPCR (Figure 4C-D) and confirmed that Fcer1a was indeed expressed by CD45+, CD117+ cardiac mast cell (Figure 4E). We then quantified the total number of cardiac mast cell after Toluidine blue staining of heart section. We found no significant differences between Lepr db/db mice and their control littermates (Figure 4F-G).

However, when we calculated the percentage of activated/degranulating mast cells, we found that it was significantly increased in the heart of Lepr db/db mice (Figure 5A-B). Since mast cells are typically activated by IgE, we measured IgE circulating levels in plasma from Lepr db/db mice and control littermates. Consistent with increased mast cell activation, IgE circulating level were significantly increased in Lepr db/db mice (Figure 5C-D). However, the percentage of IgE positive, mast cells (Fcer1a+ cells) infiltrated in the heart of Lepr db/db mice was not increased compared to that of control mice (Figure 5E) suggesting that mast cells may be activated by another pathway. Alternatively, mast cells can be activated by the complement cascade of, which we found 9 genes to be significantly increased in the transcriptomic analysis (Figure 5F).

Mast cells are known to induce cardiovascular effects through the release of their granule content [START_REF] Kritikou | The impact of mast cells on cardiovascular diseases[END_REF] , notably proteases, histamine and the production of eicosanoids and cytokines.

Consistent with increased cardiac mast cell activation, both Tryptase (Tpsab1) and Chymase (Cma1) mRNA were significantly increased in the heart of Lepr db/db mice compared to the one of their control littermates (Figure 5G-H). Also the transcriptomic analysis of the cardiac vascular fraction revealed increased eicosanoid metabolic process attested by increased arachidonate 5-lipoxygenase (Alox5), thromboxane A synthase 1 (Tbxas1) and leukotriene C4 synthase (Ltc4s) expression (Figure 5I). The increased Alox5 mRNA expression was confirmed in total heart samples via RT-qPCR (Figure 5J).

Activated cardiac mast cells, via histamine release, induce cardiac small vessel disease in

Lepr db/db mice.

To investigate the role of mast cell degranulation in the pathophysiology of cardiac microvessel disease, we first performed in vitro assays using cultured ECs. More specifically, HUVECs were co-cultured in the presence of mast cells treated with plasma from Lepr db/db or control mice. We found that HUVEC permeability was significantly increased in the presence of mast cells activated with Lepr db/db plasma compared to mast cells treated with control plasma (Supplemental Figure VIIA). Notably, the plasma of Lepr db/db mice alone was not sufficient to increase HUVEC permeability (Supplemental Figure VIIB). Since mast cells are known to promote endothelium permeability through histamine release [START_REF] Mikelis | RhoA and ROCK mediate histamine-induced vascular leakage and anaphylactic shock[END_REF] via activation of H1R receptors, we evaluated endothelial adherens junction integrity in the presence or not for cetirizine, an H1R inhibitor. We found that mast cells treated with Lepr db/db plasma increased Cdh5 junction thickness in the absence of cetirizine but not in the presence of cetirizine (Supplemental Figure VIIC-D), demonstrating that mast cells treated with Lepr db/db plasma disrupt endothelial adherens junction via histamine release, notably, we also verified that histamine alone was sufficient to alter Cdh5 junctions integrity (Supplemental Figure

VIIE-F).

To assess the role of mast cell activation in vivo in the heart, we first used cromolyn sodium, a mast stabilizer which inhibits mast cell degranulation. 2 month old Lepr db/db mice (i.e.

before they display increased EDP) were treated with 50 mg/Kg/day cromolyn sodium versus vehicle. Mice were sacrificed 28 days later. We verified cromolyn sodium therapy was To summarize, mast cells promote cardiac capillary permeability and vasodilation, which is consistent with the well-known effect of histamine contained in mast cell granules [START_REF] Ali Komi | Mast Cell Biology at Molecular Level: a Comprehensive Review[END_REF] In conclusion, mast cells promote cardiac small vessel disease via histamine release in Lepr db/db mice. Notably, cetirizine is a selective Histamine H1-receptor antagonist demonstrating that Histamine promotes cardiac capillary dilatation and leakage via Histamine H1-receptor.

Activated cardiac mast cells promote the appearance of diastolic dysfunction in Lepr db/db

mice.

Finally, to measure the pathophysiological consequences of histamine-induced small vessel disease on cardiac function, we investigated cardiac function in both cromolyn sodiumtreated and cetirizine-treated Lepr db/db mice. Ejection fraction, which is normal in Lepr db/db mice was modulated neither by cromolyn sodium treatment (Figure 7A) nor by cetirizine treatment (Figure 7B). However, EDP was significantly decreased in both cromolyn sodiumtreated (Figure 7C) and cetirizine-treated (Figure 7D) Lepr db/db mice indicating improved diastolic function. Neither Cromolyn sodium nor Cetirizine therapy did modify heart weight, LV posterior wall thickness or the mean cardiomyocyte size (Figure 7E-K), indicating that cromolyn sodium nor cetirizine therapy prevents cardiac hypertrophy.

Altogether these results indicated that mast cells, via secretion of their granule content, promote the development of diastolic dysfunction, however, they do not participate in the development of cardiomyocyte hypertrophy.

Discussion

The present study supports the microvascular hypothesis of HFpEF especially in the setting of obesity and type 2 diabetes. In this paper, we used Lepr db/db female mice as a model of diastolic dysfunction. Lepr db/db female mice have the advantage of recapitulating the main risk factors for HFpEF, i.e. diabetes, obesity female sex and hypertension. Lepr db/db mice were previously shown to display diastolic dysfunction [START_REF] Alex | Characterization of a mouse model of obesity-related fibrotic cardiomyopathy that recapitulates features of human heart failure with preserved ejection fraction[END_REF][START_REF] Pant | Genome-wide differential expression profiling of lncRNAs and mRNAs associated with early diabetic cardiomyopathy[END_REF][START_REF] Hanif | Left atrial remodeling, hypertrophy, and fibrosis in mouse models of heart failure[END_REF] and to recapitulate significant features of human HFpEF [START_REF] Horgan | Murine models of diastolic dysfunction and heart failure with preserved ejection fraction[END_REF][START_REF] Alex | Characterization of a mouse model of obesity-related fibrotic cardiomyopathy that recapitulates features of human heart failure with preserved ejection fraction[END_REF] . In the present study, we thoroughly characterized the cardiac microvascular phenotype of these mice, notably, via a transcriptomic analysis. We revealed that cardiac microvessel disease is characterized by a decreased capillary density, abnormal vessel permeability and slight constriction of arterioles but increased capillary diameter; moreover we showed that ECs display oxidative stress and have a pro-inflammatory and procoagulant phenotype. Strikingly, we demonstrated for the first time that, in Lepr db/db mice, cardiac microvessel disease is associated with increased mast cell activation and proved that it participates to the pathophysiology of both cardiac microvessel disease and diastolic dysfunction (Supplemental Figure IX). More specifically, activated mast cells release histamine which increases capillary permeability and diameter via H1 receptors known to be expressed by endothelial cells.

The current paradigm for HFpEF proposes that myocardial remodelling and dysfunction in HFpEF results from the following sequence of events: 1) comorbidities including obesity, diabetes and/or hypertension would induce a systemic low grade pro-inflammatory state;

2) this pro-inflammatory state would induce EC dysfunction characterized by an increased ROS production, a decreased NO synthesis and an increased expression of adhesion molecules such as Vcam1 and E-selectin; 3) EC dysfunction would lead to a compromised heart perfusion secondary to impaired NO-dependent vasodilatation, oedema and proinflammatory/pro-thrombotic phenotype, macrophage infiltration and fibrosis [START_REF] Marti | Endothelial dysfunction, arterial stiffness, and heart failure[END_REF][START_REF] Marechaux | Vascular and Microvascular Endothelial Function in Heart Failure With Preserved Ejection Fraction[END_REF] . The present data show that the Lepr db/db female mice model largely recapitulates this paradigm while adding further features. Notably, we demonstrated that mast cell activation, which is either part of the low grade pro-inflammatory state or induced by the low grade inflammatory state of diabetic obese mice, promotes microvascular dysfunction especially vascular permeability and capillary dilation and participates in the development of diastolic dysfunction. Consistent with a central role of inflammation [START_REF] Tromp | Identifying Pathophysiological Mechanisms in Heart Failure With Reduced Versus Preserved Ejection Fraction[END_REF] and microvascular disease in the pathophysiology of HFpEF, Lepr db/db mice do not display significant cardiac fibrosis or major cardiomyocyte abnormalities. Notably, we found that cardiomyocyte hypertrophy does not seem to promote the increased EDP. Indeed, although cromolyn sodium therapy prevents increased EDP, this effect is not associated to cardiomyocyte hypertrophy and dedifferentiation. Notably, increased EDP seen in HFpEF is not always due to cardiac hypertrophy since 31% of patients with HFpEF do not have cardiac hyperthrophy [START_REF] Shah | The many faces of heart failure with preserved ejection fraction[END_REF] . Besides cardiac hypertrophy is notably triggered by Nuclear factor of activated T-cells (NFAT) activation, NFAT is positively regulated by calmodulin/calcineurin but negatively regulated by Glycogen Synthase Kinase 3 Beta (GSK3β) [START_REF] Schirone | A Review of the Molecular Mechanisms Underlying the Development and Progression of Cardiac Remodeling[END_REF] . So cardiac hypertrophy is likely secondary to insulin resistance in type 2 diabetic mice. The present paper highlight a causal role of mast cell activation in obesity/diabetes-related increase in ventricular diastolic pressure since EDP is significantly decreased in both cromolyn and cetirizine treated mice. Considering that cromolyn and cetirizine treatment did not prevent cardiomyocyte hypertrophy, the present study suggests that the increased EDP in Lepr db/db mice is not due to cardiomyocyte hypertrophy while it might be secondary to endothelial dysfunction as suggested by Paulus et al paradigm [START_REF] Paulus | A novel paradigm for heart failure with preserved ejection fraction: comorbidities drive myocardial dysfunction and remodeling through coronary microvascular endothelial inflammation[END_REF] . The mechanism responsible for increased EDP remains to be identified.

Mast cells are immune cells that reside in the connective tissues including the myocardium.

They are characterized by the expression of c-Kit receptors and by their granules containing active mediators including proteases, notably Cma1, Tpsab1 and histamine. Mast cells may be activated by IgEs via their receptor Fcer1a, Complement factors via Toll-like receptors, IgGs or cytokines [START_REF] Hermans | Mast Cells in Cardiovascular Disease: From Bench to Bedside[END_REF] . They have been associated with several cardiovascular diseases including atherosclerosis, myocardial infarction and aneurysms [START_REF] Kritikou | The impact of mast cells on cardiovascular diseases[END_REF] , pathologies in which mast cells are contributing to the pathogenesis essentially through the release of their granule content. Importantly, circulating Tryptase was recently suggested to be a marker for cardiovascular diseases [START_REF] Mohajeri | Mast cell tryptase -Marker and maker of cardiovascular diseases[END_REF] . Moreover, mast cells have been previously involved in diastolic dysfunction induced by ovariectomy in rats [START_REF] Wang | Mast Cell Inhibition Attenuates Cardiac Remodeling and Diastolic Dysfunction in Middle-aged, Ovariectomized Fischer 344 × Brown Norway Rats[END_REF] and diabetic cardiomyopathy in streptozotocin-treated mice [START_REF] Huang | Myocardial remodeling in diabetic cardiomyopathy associated with cardiac mast cell activation[END_REF] . The present study thus confirms the significant role of mast cells in cardiovascular diseases especially heart failure. How mast cells are activated in the setting of cardiovascular diseases remains unknown [START_REF] Hermans | Mast Cells in Cardiovascular Disease: From Bench to Bedside[END_REF] . We found that, in Lepr db/db mice, increased activation of mast cells is associated with increased circulating levels of IgEs nevertheless; the amount of IgE bound cardiac to mast cells was not changed. IgE/Fcer1a is the main route of mast cell activation in allergic diseases. Interestingly, IgEs were reported to be elevated in the serum of patients with cardiovascular diseases 33 34 35 including coronary arterial disease and myocardial infarction. Consistently, asthma was shown to be related to an increased incidence of coronary heart disease, particularly in women [START_REF] Liu | Asthma and risk of coronary heart disease: A meta-analysis of cohort studies[END_REF] . More specifically, one study reported that coronary flow reserve, considered as an early marker of endothelial dysfunction is significantly lower in patients with high IgE levels [START_REF] Unal | Impact of high serum Immunoglobulin E levels on the risk of atherosclerosis in humans[END_REF] . Altogether, these results further support that Lepr db/db mice are a relevant model of human cardiovascular diseases.

In conclusion, the present study further confirms that inflammation and cardiac microvessel disease are at the heart of HFpEF pathophysiology [START_REF] Tromp | Identifying Pathophysiological Mechanisms in Heart Failure With Reduced Versus Preserved Ejection Fraction[END_REF] and identified for the first time mast cells as critical players of cardiac microvessel disease and diastolic dysfunction, making them a promising therapeutic target for HFpEF treatment. Indeed the present study suggests that mast cell stabilizers are H1 receptor antagonists may be used to prevent the occurrence of heart failure. Notably, while H2 receptor antagonist are widely suggested to have beneficial effects in heart failure especially because of their negative inotropic and chronotropic effects [START_REF] Zhang | Icosanoid metabolic process" is one of the molecular function significantly increased in Lepr db/db mice (n=4 mice/group). (J) Alox5 mRNA expression was measured via RT-qPCR in total heart biopsies (n=10 mice/group)[END_REF] . The cardiac effects of H1 receptor antagonists remain poorly investigated. Interestingly, one study reported that H1 receptor antagonists may improve cardiac remodeling in patients on chronic hemodialysis 39 . Another study performed in mice reported that cetirizine improves cardiac remodeling associated with viral myocarditis 40 .

This work benefited from equipment and services from the iGenSeq (RNA sequencing) and iCONICS (RNAseq analysis) core facilities at the ICM (Institut du Cerveau et de la Moelle épinière, Hôpital Pitié-Salpêtrière, PARIS, France).

Sources of Funding

This study was supported by grants from the Fondation pour la Recherche Médicale (équipe

Figures and

  Lepr db/db mice and their control littermates (Supplemental FigureIIIA-B). Then we performed RT-qPCR to measure expression level of several cardiomyocyte markers notably, Titin isoforms N2-A and N2-B, Myocyte-specific enhancer factor 2C (MEF2C), Cx43, Sarcoplasmic/endoplasmic reticulum calcium ATPase 2 (Serca2) and Cardiac phospholamban (Pln) (Supplemental FigureIIIC-H). None of these markers were significantly different in Lepr db/db mice compared to their control littermates.

  and in their control littermates (Supplemental Figure IVA-B). This result was confirmed using the Sircol™ collagen assay (Supplemental Figure IVC). Moreover collagen type I alpha 1 chain (Col1a1) mRNA expression was not modified, while collagen type III alpha 1 chain (Col3a1) mRNA expression was diminished (Supplemental Figure IV-E). Finally, no difference in Lysyl oxidase (Lox) mRNA expression, an enzyme promoting collagen cross linking was observed (Supplemental Figure IVF). To test whether cardiac fibrosis may appear later in life, we performed the same assays in a year old animals. Similarly to what we found in 3 month old animal, neither picrosirius red stain nor the Sircol™ collagen assay showed any differences between Lepr db/db mice and control littermates (Supplemental Figure IVG-H). Although Lox mRNA was significantly increased in a year old Lepr db/db mice compared to their control littermates (Supplemental Figure IVJ), it did not lead to increased amount of insoluble collagen in the heart (Supplemental Figure IVK).

  ). Accordingly, transcriptomic analysis via RNA sequencing of the cardiac vascular fraction of these mice showed a switch in the balance between vasodilator and vasoconstrictor molecules in favor of vasoconstriction of cardiac arterioles (Figure2F) notably, because of the downregulated expression of Apelin (Apln) and the increased expression of Type-1 angiotensin II receptor (Agtr1), Thromboxane-A synthase (Tbxas1), Arachidonate 5-lipoxygenase (Alox5) and Prostaglandin G/H synthase 2 (Ptgs2) (Figure2F). However, Nos3 and phospho-Nos3 levels were not different in Lepr db/db and control mice (Figure2G-I).To assess vascular permeability we measure albumin extravasation and found a significantly increased albumin extravasation in Lepr db/db mice compared to their control littermates which indicate abnormal vessel permeability (Figure2J-K). Besides, both the transcriptomic analysis and gene expression analysis showed a significant upregulation of multiple inflammatory cytokines including Il-6, chemokine (C-C motif) ligand 2 (Ccl2), chemokine C-C motif 3 ligand (Ccl3) and Ccl7 (Figure3A-D). Il-6 and Ccl7 overexpression were confirmed at protein level (Figure3D-F). Also, we found an increased mRNA expression of adhesion molecule Icam1 and E-selectin (Supplemental FigureVC-F). However, the increased Icam1 expression was not confirmed at protein level (Supplemental Figure VG-I). The transcriptomic analysis also revealed a pro-coagulation phenotype attested by P2Y purinoceptor 12 (P2ry12), Coagulation factor XIII A chain (F13a1) and Platelet factor 4 (PF4) upregulation (Supplemental Figure VJ). Finally, we assessed oxidative stress which was not different in Lepr db/db and control mice via DHE staining (Supplemental Figure VA-B).

  Leucocytes, cells were mostly CD68+/Mrc1+ M2 macrophages (Supplemental Figure VIA-B). Besides, leucocytes also included CD3+ T-lymphocytes (Supplemental Figure VIC-D), Ly6G+ neutrophils (Supplemental Figure VIE-F) and B220 B-lymphocytes (Supplemental Figure VIG-H). Both M2 macrophage and B-lymphocyte densities were increased in Lepr db/db mice.

  effective and did decrease the percentage of degranulating mast cells in the heart (Supplemental Figure VIIIA). Then, we investigated the effect of cromolyn sodium therapy on cardiac microvessel phenotype and cardiac inflammation. Cromolyn sodium therapy did not modify cardiac microvessel density (Supplemental Figure VIIIB), however it did reduce capillary diameter (Figure 6A-B) and permeability attested by decreased albumin extravasation (Figure 6C-D). Moreover, CD45+ leucocyte, especially CD68+ macrophage, recruitment was significantly decreased (Figure 6E-F, Supplemental Figure VIIIC-F).

  . To verify cardiac capillary permeability and vasodilation was indeed due to histamine, 2 month old Lepr db/db mice were treated with 4 mg/kg/day cetirizine versus vehicle. As expected, both the diameter of cardiac capillaries (Figure 6G-H) and their permeability (Figure 6I-IJ) were significantly reduced in Lepr db/db mice treated with cetirizine vs vehicle-treated Lepr db/db mice. Also, similarly, to cromolyn sodium therapy, cetirizine therapy significantly prevented CD68+ macrophage recruitment (Figure 6K-L, Supplemental Figure VIIIG-J).

FRM), fromHighlights

  the Region « Nouvelle Aquitaine » and the Agence Nationale pour la Recherche (Appel à Projet Générique). Finally, this study was co-funded by the "Institut National de la Santé et de la Recherche Médicale" and by the University of Bordeaux. Disclosures None declared 39. Omae K, Ogawa T, Yoshikawa M, Nitta K. The use of H1-receptor antagonists and left ventricular remodeling in patients on chronic hemodialysis. Heart Vessels. 2010;25(2):163-169. doi:10.1007/s00380-009-1183-9 40. Matsumori A, Yamamoto K, Shimada M. Cetirizine a histamine H1 receptor antagonist improves viral myocarditis. J Inflamm (Lond). 2010;7:39. doi:10.1186/1476-9255-7-39 Lepr db/db female display increased cardiac mast cell activation  Cardiac mast cell activation participates in the development of cardiac small vessel disease, notably via Histamine release.  Cardiac mast cell activation participate in the development of diastolic dysfunction  Mast cell stabilizer or H1 receptor antagonist may be used to prevent the occurrence of diastolic dysfunction in the setting of type diabetes and obesity.

Figure 1 :

 1 Figure 1: Lepr db/db female mice have diastolic dysfunction from 3 months of age. Lepr db/db female mice and their control Lepr db /+ littermates were subjected to echocardiography, LV

Figure 2 :

 2 Figure 2: Lepr db/db female mice have cardiac microvascular dysfunction. Lepr db/db female mice and their control Lepr db /+ littermates were sacrificed at 3 months of age. (A) Heart

Figure 3 :

 3 Figure 3: Lepr db/db female mice have increased cardiac inflammation. Lepr db/db female mice and their control Lepr db /+ littermates were sacrificed at 3 months of age. (A) RNA

Figure 4 :

 4 Figure 4: Lepr db/db female mice have increased cardiac inflammation. Lepr db/db female mice and their control Lepr db /+ littermates were sacrificed at 3 months of age. (A) RNA

Figure 5 :

 5 Figure 5: Lepr db/db female mice have increased cardiac mast cell activation. Lepr db/db female mice and their control Lepr db /+ littermates were sacrificed at 3 months of age. (A) Heart

Figure 6 :

 6 Figure 6: Mast cells promote cardiac small vessel disease in Lepr db/db female mice. (A-F) 2 month old Lepr db/db female mice were treated or not with 50 mg/kg/day cromolyn sodium

Figure 7 :

 7 Figure 7: Mast cells promote diastolic dysfunction in Lepr db/db female mice. 2 month old Lepr db/db female mice were either treated with 50 mg/kg/day cromolyn sodium vs control

  db/db female mice are significantly overweight from 6 weeks of age while diabetes appears from 3 months of age. Plasmatic HDL cholesterol is elevated from 6 weeks of age while LDL cholesterol is only increased after 6 months of age. Plasmatic triglycerides

are elevated from 3 months of age. Besides Lepr db/db female mice have mild hypertension from 3 months of age and display significantly increased circulating leucocytes (monocytes and neutrophils) later in life at 1 year of age (Supplemental Figure

I

).

Acknowledgments

We thank Philippe Alzieu, Annabel Reynaud, Sylvie Jeanningros, Sylvain Grolleau, and Maxime David for their technical help. We thank Christelle Boullé for administrative assistance.