V Santhosh
email: santhosh.priya@maths.christuniversity.in2sudev.nk@christuniversity.in

Priya 𝟏

N K Sudev 𝟐

INJECTIVE COLORING PARAMETERS OF SOME SPECIAL CLASSES OF GRAPHS

Keywords: Coloring mean, coloring variance, 𝜉 -chromatic mean, 𝜉chromatic variance, 𝜉 + -chromatic mean, 𝜉 + -chromatic variance 05C15, 05C38

The vertex coloring of a graph can be viewed as a random experiment and with respect to this experiment, a random variable 𝑋 can be defined such that 𝑋 denotes the color of an arbitrarily chosen vertex. In this paper, we broaden the ideas of coloring mean and variance of graphs with respect to one special type of proper injective coloring and determine these parameters for some standard graph classes.

Introduction

We follow [START_REF] Bondy | Graph theory[END_REF][START_REF] Harary | Graph theory[END_REF][START_REF] E W Weisstein | CRC concise encyclopedia of mathematics[END_REF][START_REF] West | Introduction to graph theory[END_REF] all the terms and definitions, not defined specifically in this paper. Moreover, for notions and norms in graph coloring, see [START_REF] Chartrand | Chromatic graph theory[END_REF][START_REF] Jensen | Graph coloring problems[END_REF][START_REF] Kubale | Graph colorings[END_REF]. All graphs considered here are undirected, simple, finite and connected.

A graph coloring is a mapping 𝑐: 𝑉(𝐺) → 𝐶 = {𝑐 1 , 𝑐 2 , … , , 𝑐 𝑙 }; a set of colors, labels or weights. Unless specified otherwise, by a graph coloring we mean an assignment of colors to the vertices. A coloring of a graph in which adjacent vertices get didtincct colors is known as a proper coloring. For many decades, graph coloring techniques have turned out to be useful tools in modelling the theoretical and practical problems.

A few graph coloring parameters, identified with various kinds of graph coloring have been presented in the literature and considered their properties broadly. The cardinality of a minimal proper coloring of the given graph is called its chromatic number which remains to be the most studied parameter in this area. Afterwards, the chromatic sum of graphs as for different graph colorings have been presented and considered by numerous researchers. A portion of these studies can be found in (see [START_REF] K P Chithra | On equitable coloring parameters of certain cycle related graphs[END_REF][START_REF] Hahn | On the injective chromatic number of graphs[END_REF][START_REF] Kishore | Injective chromatic sum and injective chromatic polynomials of graphs[END_REF][START_REF] Sudev | Certain chromatic sums of some cycle-related graph classes[END_REF][START_REF] Sudev | A study on the injective coloring parameters of certain graphs[END_REF][START_REF] Sudev | On certain parameters of equitable coloring of graphs[END_REF][START_REF] Sudev | On certain colouring parameters of graphs[END_REF]). The idea of a general coloring sum of a graph has been discussed in (see [START_REF] Kubicka | An introduction to chromatic sums[END_REF][START_REF] Kok | Generalised colouring sums of graphs[END_REF]) as pursues.

Let 𝐶={𝑐 1 , 𝑐 2 𝑐 3 , … , 𝑐 𝑘 } be a specific kind of appropriate 𝑘-color set of any given graph and 𝜃(𝑐 𝑖 ) indicates the number of occurences a specific color 𝑐 𝑖 in the coloring of 𝐺. For a coloring 𝐶 the corresponding coloring sum for any given graph is indicated by 𝜔 𝐶 (𝐺) is determined to be 𝜔 𝒞 (𝐺) = ∑ 𝑘 𝑖=1 𝑖𝜃(𝑐 𝑖 ) (see [START_REF] Kok | Generalised colouring sums of graphs[END_REF]).

Any proper graph coloring of can be treated as a random experiment, where the corresponding random variable 𝑋 denotes the color of an arbitrarily chosen vertex. The weight of a graph 𝐺 is the sum of all weights of colors of 𝐺 and the probability mass function (𝑝. 𝑚. 𝑓) of 𝑋 valued function 𝑓(𝑖) is defined in [START_REF] Sudev | On certain colouring parameters of graphs[END_REF] as

𝑓(𝑖) = ( 𝜃(𝑐 𝑖 ) |𝑉(𝐺)| , 𝑖𝑓 𝑖 = 1,2,3, … , 𝑘 ,

0; 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

If the context is clear, we can say that 𝑓(𝑖) is the 𝑝. 𝑚. 𝑓 of the corresponding graph 𝐺 with respect to the given colouring 𝐶.

The mean and variance of a graph 𝐺, concerning a general coloring of 𝐺 are characterized in [START_REF] Sudev | On certain colouring parameters of graphs[END_REF] as follows.

Let 𝐶={𝑐 1 , 𝑐 2 𝑐 3 , … , 𝑐 𝑘 } be a proper 𝑘 color set for any given graph 𝐺. Then, the coloring mean of a coloring 𝐶 of a given graph 𝐺, denoted by 𝜇 𝐶 (𝐺), is given by

𝜇 𝐶 (𝐺) = ∑ 𝑘 𝑖=1 𝑖𝑓(𝑖) = ∑ 𝑘 𝑖=1 𝑖𝜃(𝐶 𝑖 ) ∑ 𝑘 𝑖=1 𝜃(𝐶 𝑖 )
.

For a positive integer 𝑟, the 𝑟-the moment of the coloring 𝐶 is denoted by 𝜇 𝐶 𝑟 (𝐺) is defined as

𝜇 𝐶 𝑟 (𝐺) = ∑ 𝑘 𝑖=1 𝑖 𝑟 𝑓(𝑖).
The coloring variance of a coloring 𝐶 of a given graph 𝐺, denoted by 𝜎 𝐶 2 (𝐺), is defined to be

𝜎 𝐶 2 (𝐺) = 𝜇 𝐶 2 (𝐺) -(𝜇 𝐶 (𝐺)) 2 .
Motivated by the above mentioned studies, in this paper, we broaden the notions of coloring mean and variance to a modified injective coloring of graphs. Definition 2.5 [START_REF] Sudev | A study on the injective coloring parameters of certain graphs[END_REF] The 𝜉 + -chromatic variance of a graph 𝐺 indicated by 𝜎 𝜉 + 2 (𝐺) is a coloring variance of 𝐺 in accordance with minimal injective coloring 𝐶 of 𝐺 such that the coloring sum is maximum.

In the following discussion, we discuss these injective chromatic parameters for some fundamental graph classes.

The sunlet graph 𝑆𝑙 𝑛 is a graph obtained by attaching a pendant edge to every vertex of a cycle. That is, 𝑆𝑙 𝑛 = 𝑐 𝑛 ⊙ 𝐾 1 , the corona of a cycle 𝐶 𝑛 and 𝑘 1 .Note that when 𝑛 = 5, the 𝜉chromatic mean and 𝜉 -chromatic variance of the sunlet graph 𝑆𝑙 𝑛 is given by and the 𝜉-chromatic variance of the sunlet graph when 𝑛 ≠ 5 is

𝜎 𝜉 2 (𝑆𝑙 𝑛 ) = ( 4 3 , 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 3) , 4𝑛 2 +2𝑛-12 3𝑛 2 , 𝑖𝑓 𝑛 ≡ 1 (𝑚𝑜𝑑 3) , 4𝑛 2 +13𝑛-27 3𝑛 2
, 𝑖𝑓 𝑛 ≡ 2 (𝑚𝑜𝑑 3) .

Proof. The vertices of the cycle are given by V={𝑣 1 , 𝑣 2 , 𝑣 3 , … , 𝑣 𝑛 } and the vertices of pendent edges are given by 𝑈 = {𝑢 1 , 𝑢 2 , 𝑢 3 , … , 𝑢 𝑛 }. The, we need to consider the following cases.

Case 1: Let 𝑛 ≡ 0(𝑚𝑜𝑑 3) . Then, the 𝜉 -chromatic number for sunlet graph is 4 that is 𝜉(𝑆𝑙 𝑛 )=4thus, we require 4 colors say 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 (See Figure 1). , 𝑖𝑓 𝑖 = 2,3,4 , 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 . Thus, the 𝜉 -chromatic mean is given by 𝜇 𝜉 (𝑆𝑙 𝑛 )=

1 2 + (2 + 3 + 4) 1 6 =2 and the 𝜉 -chromatic variance is 𝜎 𝜉 2 (𝑆𝑙 𝑛 )=( 1 2 )2 + (2 2 + 3 2 + 4 2 )( 1 6
)-(𝜇 𝜉 (𝑆𝑙 𝑛 )) 2 =

chromatic variance is given by 𝜎 𝜉 2 (𝑆𝑙 𝑛 ) = ( 𝑛-2 2𝑛 + (2 2 + 3 2 + 4 2 ) 𝑛+2 6𝑛 ) = (𝜇 𝜉 (𝑆𝑙 𝑛 )) 2 = 4𝑛 2 +2𝑛-12 3𝑛 2
.

Case 3: Let 𝑛 ≡ 2(𝑚𝑜𝑑 3). In this case, the chromatic number is 5 that is 𝜉(𝑆𝑙 𝑛 ) = 5. Thus, we need 5 colors, say 𝑐 1 , 𝑐 3.

Figure 3

Here, the corresponding 𝑝. 𝑚. 𝑓 is

𝑓(𝑖) = ( 𝑛-2 2𝑛 , 𝑖𝑓 𝑖 = 1 , 𝑛+1 6𝑛 , 𝑖𝑓 𝑖 = 2,3, 4 , 1 2𝑛
, 𝑖𝑓 𝑖 = 5 , 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 .

Thus, the 𝜉-chromatic mean is given by 𝜇 𝜉 (𝑆𝑙 𝑛 )= ) -

(𝜇 𝜉 (𝑆𝑙 𝑛 )) 2 = 4𝑛 2 +13𝑛-27 3𝑛 2
.

Note that when 𝑛 = 5, the 𝜉 + -chromatic mean and the 𝜉 + -chromatic variance of the sunlet graph 𝑆𝑙 𝑛 is given by respectively. Then, we will discuss the 𝜉 + -chromatic mean and variance of sunlet graph, with 𝑛 ≥ 5.

Theorem 2.2

The 𝜉 + -chromatic mean of a sunlet graph when 𝑛 ≠ 5 is

𝜇 𝜉 + (𝑆𝑙 𝑛 ) = ( 3, 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 3) 3𝑛-2 𝑛 , 𝑖𝑓 𝑛 ≡ 1 (𝑚𝑜𝑑 3) 4𝑛-3 𝑛
, 𝑖𝑓 𝑛 ≡ 2 (𝑚𝑜𝑑 3) . and the 𝜉 + -chromatic variance of the sunlet graph, when 𝑛 ≠ 5, is given by

𝜎 𝜉 + 2 (𝑆𝑙 𝑛 ) = ( 4 3
, 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 3)

4𝑛 2 +2𝑛-12 3𝑛 2 , 𝑖𝑓 𝑛 ≡ 1 (𝑚𝑜𝑑 3) 4𝑛 2 +13𝑛-27 3𝑛 2
, 𝑖𝑓 𝑛 ≡ 2 (𝑚𝑜𝑑 3) .

Proof. In this context, we need to consider the following cases.

Case 1: Let 𝑛 ≡ 0(𝑚𝑜𝑑 3). The 𝜉 + -coloring of 𝑆𝑙 𝑛 in this case is depicted in 4.

Figure 4

The 𝜉 + -chromatic number for sunlet graph is , 𝑤ℎ𝑒𝑛 𝑖 = 1,2,3

, 𝑤ℎ𝑒𝑛 𝑖 = 4 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 .

Thus, the 𝜉 + -chromatic mean is given by 𝜇 𝜉 + (𝑆𝑙 𝑛 ) = (1 + 2 + 3+) .

Case 3: Let 𝑛 ≡ 2(𝑚𝑜𝑑 3). In this case, an injective coloring of a sunlet graph 𝑆𝑙 𝑛 is illustrated in Figure 6.

Figure 6

Here, the chromatic number is 5 that is 𝜉 + (𝑆𝑙 𝑛 )=5. Thus, we need 5 colors say 𝑐 1 , 𝑐 , 𝑖𝑓 𝑖 = 5 .

0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 .

Thus, the 𝜉 + -chromatic mean is given by 𝜇 𝜉 + (𝑆𝑙 𝑛 ) = The 𝜉 + -coloring of a the sunlet graph using 4 colors is shown in Figure 7.

Figure 7

Then, the corresponding 𝑝. 𝑚. 𝑓 is

𝑓(𝑖) = ( ⌈ 𝑛 2 ⌉ 2𝑛 , 𝑖𝑓 𝑖 = 1, 2 , ⌊ 𝑛 2 ⌋ 2𝑛
, 𝑖𝑓 𝑖 =

Thus, the 𝜉-chromatic mean is given by 𝜇 𝜉 (𝐶𝐿 𝑛 ) = (1 + 2) .

Case 2: Let 𝑛 be an odd number. The 𝜉 + -coloring of the graph is shown in Figure 8. The 𝜉 + -chromatic variance of a closed ladder graph is

𝜎 𝜉 + 2 (𝐶𝐿 𝑛 ) = ( 5𝑛 2 -4 2𝑛 , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 , 5 4
, 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 .

Proof. The vertex set of path 𝑝 𝑛 1 is given by 𝑉={𝑣 1 , 𝑣 2 , 𝑣 3 …} and the vertex set of 𝑝 𝑛 2 is given by 𝑈={𝑢 1 , 𝑢 2 , 𝑢 3 …}. Here, the 𝜉 + -chromatic number is 4 that is 𝜉 + (𝐶𝐿 𝑛 ) = 4. Thus, we need 4 color classes say 𝑐 1 , 𝑐 2 , 𝑐 3 , and 𝑐 4 .

Case 1: Let 𝑛 be an even number. The corresponding coloring of the closed ladder is depicted in Figure 9.

Figure 9

The corresponding color classes for 𝑐 1 , 𝑐 .

Case 2: Let𝑛 be an odd number. Then, the required chromatic coloring is shown in Figure10.

Figure 10

The corresponding color classes for 𝑐 1 , 𝑐 Then, the corresponding 𝑝. 𝑚. 𝑓 is given by

𝑓(𝑖) = ( ⌊ 𝑛 2 ⌋ 2𝑛 , 𝑖𝑓 𝑖 = 1,2 , ⌈ 𝑛 2 ⌉ 2𝑛
, 𝑖𝑓 𝑖 = 3,4 , 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 .

Thus, the 𝜉 + -chromatic mean is given by 𝜇 𝜉 + (𝐶𝐿 𝑛 ) = (1 + 2) A helm graph is obtained by adding pendent edges to the rim vertices of a wheel. When 𝑛 = 3 the 𝜉-chromatic mean of a helm graph is .

For 𝑛 = 3, the 𝜉 + -chromatic mean of a helm graph is Proof. Since the central vertex is adjacent to all other vertices of the rim 𝑛 + 1 colors are required to color the graph. That is, 𝜉 + (𝐻 𝑛 ) = 𝑛 + 1. The pendant vertices receives the least color missed by the vertices at a distance 2 from it.

Then, the corresponding 𝑝. 𝑚. 𝑓 is

Theorem 2 . 1

 21 injective chromatic parameters of sunlet graphs 𝑆𝑙 𝑛 with 𝑛 ≥ 6, have been determined in the following result. The 𝜉-chromatic mean of a sunlet graph, when 𝑛 ≠ 5 is 𝜇 𝜉 (𝑆𝑙 𝑛 )

Figure 1 Then

 1 Figure 1

4 3 . 2 :

 432 Case Let 𝑛 ≡ 1(𝑚𝑜𝑑 3) In this case also, the 𝜉-chromatic number remains the same that is 𝜉(𝑆𝑙 𝑛 ) = 4 . Thus, we need 4 colors say 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 . Then, the corresponding color classes for 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 are given by 𝑉 𝑐 1 = {𝑣 𝑛 , 𝑢 2 , 𝑢 3 , … 𝑢 𝑛-2 }, 𝑉 𝑐 2 = {𝑢 𝑛 , 𝑣 2 , 𝑣 5 , 𝑣 8 , … }, 𝑉 𝑐 3 = {𝑢 1 , 𝑣 3 , 𝑣 6 , 𝑣 9 , … }, and 𝑉 𝑐 4 = {𝑢 𝑛-1 , 𝑣 1 , 𝑣 4 , 𝑣 7 , … } (See Figure2).

Figure 2

 2 Figure 2

  -chromatic variance is given by 𝜎 𝜉 2 (𝑆𝑙 𝑛 ) = (

4

 4 and thus, we require 4 colors say 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 . Then, the corresponding color classes for 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 are given by 𝑉 𝑐 1 = {𝑣 3 , 𝑣 6 , … , 𝑣 𝑛 }, 𝑉 𝑐 2 = {𝑣 2 , 𝑣 5 , … , 𝑣 𝑛-1 }, 𝑉 𝑐 3 = {𝑣 1 , 𝑣 4 , … , 𝑣 𝑛-2 } and 𝑉 𝑐 4 = {𝑢 1 , 𝑢 2 , … , 𝑢 𝑛 } and the corresponding 𝑝. 𝑚. 𝑓 is 𝑓(𝑖) = ( 1 6

Figure 5

 5 Figure 5

Figure 8 Theorem 2 . 4

 824 Figure 8

  𝜉 + -chromatic mean is given by 𝜇 𝜉 + (𝐶𝐿 𝑛 )=(1

.Theorem 2 . 5 2 Proof.

 252 Then, the following theorem deals with the 𝜉-chromatic mean and variance of helm graphs, where 𝑛 ≥ 4. For 𝑛 > 3, the 𝜉-chromatic mean of a helm graph is𝜇 𝜉 (𝐻 𝑛 ) = 𝑛 2 +5𝑛+144𝑛+2and the 𝜉-chromatic variance of the helm graph is Since the central vertex is adjacent to all other vertices of the rim 𝑛 + 1 colors are required to color the graph that is 𝜉 ( 𝐻 𝑛 ) = 𝑛 + 1. The pendant vertices receives the least color missed by the vertices at a distance 2 from it. Then their 𝑝. 𝑚. 𝑓 is given by 𝑓(𝜉 -chromatic mean is given by 𝜇 𝜉 (𝐻 𝑛 ) =

25 7 andTheorem 2 . 6

 726 the 𝜉 + -chromatic variance of the helm graph is 𝜎 𝜉 + 2 (𝐻 𝑛 ) = 110 49. Then, the following theorem discusses the 𝜉 + -chromatic mean of a helm graph 𝐻 𝑛 with 𝑛 ≥ 4. For 𝑛 > 3, the 𝜉 + -chromatic mean of a helm graph is

  The idea of injective coloring in graph theory have gained interest of various researchers and many intriguing investigations have been done here. In this paper, unique in relation to the current working on injective coloring we utilize the slightly modified definition of injective coloring and the results are obtained as follows. The 𝜉-chromatic mean or injective chromatic mean of a graph 𝐺 indicated by 𝜇 𝜉 (𝐺) is a coloring mean of 𝐺 with minimum proper injective coloring 𝐶 of 𝐺 and minimum coloring sum𝜔 𝐺 .

	Definition 2.1 [12]A proper injective coloring 𝐶 of graph is a proper coloring 𝐶 in which
	neighbours of same vertex should have different colors.
	The chromatic mean and variance of graph 𝐺 in accordance with injective coloring is defined as
	follows:
	Definition 2.2 [12] Definition 2.3 [12] The 𝜉-chromatic variance of a graph 𝐺, denoted by 𝜎 𝜉 2 (𝐺) is a coloring
	variance of 𝐺 in accordance with minimal injective coloring 𝐶 of 𝐺 such that the coloring
	sum is minimum.
	Definition 2.4 [12] The 𝜉 + -chromatic mean or injective chromatic mean of a graph 𝐺
	indicated by 𝜇 𝜉 + (𝐺) is a coloring mean of 𝐺 with minimum proper injective coloring 𝐶 of 𝐺
	and maximum coloring sum𝜔 𝐺 .

  2 , 𝑐 3 , 𝑐 4 and 𝑐 5 , to color the vertices of the graph. Then, the corresponding color classes for 𝑐 1 , 𝑐 2 , 𝑐 3 , 𝑐 4 and 𝑐 5 are given by 𝑉 𝑐 1 = {𝑣 𝑛-1 , 𝑢 1 , 𝑢 2 , … 𝑢 𝑛-3 }, 𝑉 𝑐 2 = {𝑢 𝑛-1 , 𝑣 2 , 𝑣 5 , … 𝑣 𝑛-3 } , 𝑉 𝑐 3 = {𝑢 𝑛 , 𝑣 3 , 𝑣 6 , … 𝑣 𝑛-2 } , 𝑉 𝑐 4 = {𝑢 𝑛-2 , 𝑣 1 , 𝑣 4 , … 𝑣 𝑛-4 } , and 𝑉 𝑐 5 = {𝑣 𝑛 }. This case is illustrated in Figure

  Here, the 𝜉 + -chromatic number remains the same that is 𝜉(𝑆𝑙 𝑛 ) = 4. Thus, we need 4 colors say 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝐶 4 . Then, the corresponding color classes for 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 are given by 𝑉 𝑐 1 = {𝑢 𝑛-1 , 𝑣 1 , 𝑣 4 , 𝑣 7 , … } , 𝑉 𝑐 2 = {𝑢 1 , 𝑣 3 , 𝑣 6 , 𝑣 9 , … } , 𝑉 𝑐 3 = {𝑢 𝑛 , 𝑣 2 , 𝑣 5 , 𝑣 8 , … } 𝑉 𝑐 4 = {𝑣 𝑛 , 𝑢 2 , 𝑢

	chromatic variance is 𝜎 𝜉 + 2 (𝑆𝑙 𝑛 ) = ((1 2 + 2 2 + 3 2 +)	1 6	+ 4 2 (	1 6 )) = (𝜇 𝜉 + (𝑆𝑙 𝑛 )) 2 = + 4( 1 ) = 3 and the 𝜉 + -2 1 2 4 3 .
	Case 2: Let 𝑛 ≡ 1(𝑚𝑜𝑑 3).

[START_REF] K P Chithra | On equitable coloring parameters of certain cycle related graphs[END_REF]

, … 𝑢 𝑛-2 }. The 𝜉 + -coloring of 𝑆𝑙 𝑛 in this case is illustrated in Figure

5

.

  2 , 𝑐 3 , 𝑐 4 and 𝑐 5 . Then, the corresponding color classes for 𝑐 1 , 𝑐 2 , 𝑐 3 , 𝑐 4 and 𝑐 5 are 𝑉 𝑐 1 = {𝑣 𝑛 }, 𝑉 𝑐 2 = {𝑢 𝑛-1 , 𝑣 2 , 𝑣 5 , … 𝑣 𝑛-3 } 𝑉 𝑐 3 = {𝑢 𝑛 , 𝑣 3 , 𝑣 6 , … 𝑣 𝑛-2 }, 𝑉 𝑐 4 = {𝑢 𝑛-2 , 𝑣 1 , 𝑣 4 , … 𝑣 𝑛-4 } 𝑉 𝑐 5 = {𝑣 𝑛-1 , 𝑢 1 , 𝑢 2 , … 𝑢 𝑛-3 } and their 𝑝. 𝑚. 𝑓 is

		1 2𝑛	,	𝑖𝑓 𝑖 = 1 .
	𝑓(𝑖) =	𝑛+1 6𝑛 𝑛-2	, 𝑖𝑓 𝑖 = 2,3,4 .
		2𝑛
	(

  Proof. The vertex set of path 𝑝 𝑛 1 is given by 𝑉={𝑣 1 , 𝑣 2 , 𝑣 3 …} and the vertex set of 𝑝 𝑛 2 is given by 𝑈={𝑢 1 , 𝑢 2 , 𝑢 3 …}. Here, the 𝜉-chromatic number is 4 that is 𝜉(𝐶𝐿 𝑛 ) = 4. Thus, we need 4 color classes say 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 . Then, we have Case 1: Let 𝑛 be an even number. The corresponding color classes for 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 are given by 𝑉 𝑐 1 = {𝑣 3 , 𝑣 7 , 𝑣 11 … , 𝑢 1 , 𝑢 5 , 𝑢 9 , … }, 𝑉 𝑐 2 = {𝑣 1 , 𝑣 5 , 𝑣 9 , … , 𝑢 3 , 𝑢 7 , 𝑢 11 , … } 𝑉 𝑐 3 = {𝑣 4 , 𝑣 8 , 𝑣 12 … , 𝑢 2 , 𝑢 6 , 𝑢 10 , … } 𝑉 𝑐 4 = {𝑣 2 , 𝑣 6 , 𝑣 10 … , 𝑢 4 , 𝑢 8 , 𝑢 12 , … }.

	1 2𝑛 and the 𝜉 + -chromatic variance is given by 𝜎 𝜉 + + (2 + 3 + 4)( 2 (𝑆𝑙 𝑛 ) = ( 1 2𝑛 + (2 2 + 3 2 + 4 2 )( 𝑛+1 6𝑛 ) + 5( 𝑛-2 2𝑛 ) = 𝑛+1 (4𝑛-3) 𝑛 ) + 6𝑛 5 2 ( 𝑛-2 2𝑛 4𝑛 2 +13𝑛-27 )) -(𝜇 𝜉 + (𝑆𝑙 𝑛 )) 2 = 3𝑛 2
	5𝑛-2 2𝑛 𝜇 𝜉 (𝐶𝐿 𝑛 ) = ( 5 2 ,	, 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 .
	The 𝜉-chromatic variance of a closed ladder graph is
	𝜎 𝜉 2 (𝐶𝐿 𝑛 ) = (	5𝑛 2 -4 2𝑛 5 4 ,	, 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 .

.

In the following theorem, 𝜉-chromatic mean and and variance of ladder graph will be discussed.

A closed ladder graph is defined as the Cartesian product of paths 𝑝 𝑛 and 𝑝 2 (i.e) 𝑝 𝑛 ◻ 𝑝 2 , Theorem 2.3 For 𝑛 ≥ 3, the 𝜉-chromatic mean of a closed ladder graph is

  2 , 𝑐 3 and 𝑐 4 are given by 𝑉 𝑐 1 = {𝑣 3 , 𝑣 7 , 𝑣 11 … , 𝑢 1 , 𝑢 5 , 𝑢 9 , … }, 𝑉 𝑐 2 = {𝑣 1 , 𝑣 5 , 𝑣 9 , … , 𝑢 3 , 𝑢 7 , 𝑢 11 , … } 𝑉 𝑐 3 = {𝑣 4 , 𝑣 8 , 𝑣 12 … , 𝑢 2 , 𝑢 6 , 𝑢 10 , … } 𝑉 𝑐 4 = {𝑣 2 , 𝑣 6 , 𝑣 10 … , 𝑢 4 , 𝑢 8 , 𝑢 12 , … }.

	Then the corresponding 𝑝. 𝑚. 𝑓 is given by
	𝑓(𝑖) =

  2 , 𝑐 3 and 𝑐 4 are given by 𝑉 𝑐 1 = {𝑣 4 , 𝑣 8 , 𝑣 12 … , 𝑢 2 , 𝑢 6 , 𝑢 10 , , … }, 𝑉 𝑐 2 = {𝑣 2 , 𝑣 6 , 𝑣 10 … , 𝑢 4 , 𝑢 8 , 𝑢 12 , … } 𝑉 𝑐 3 = {𝑣 1 , 𝑣 5 , 𝑣 9 , … , 𝑢 3 , 𝑢 7 , 𝑢 11 , … }, 𝑉 𝑐 4 = {𝑣 3 , 𝑣 7 , 𝑣 11 … , 𝑢 1 , 𝑢 5 , 𝑢 9 , … }.

Conclusion

In this paper, we discussed the statistical parameters, the mean and variance of certain classes of graphs, with respect to a particular type of injective coloring. Further investigations are possible in this direction as many graph classes are yet to be discussed in context of these coloring parameters. These parameters seem to be useful in many real life or practical problems. These can be used in many other areas like optimisations, networks, distributions etc. With the conditions related to proper injective coloring.

.

The closed helm graph is a graph obtained by adding edges between the pendent vertices in helm graph.

If 𝑛 = 3, it follows uniform distribution and thus the 𝜉-chromatic mean of a closed Helm graph is𝜇 𝜉 (𝐶𝐻 𝑛 ) = 𝑛 + 1 and the 𝜉 -chromatic variance of the closed helm graph is

. When 𝑛 = 4,5,6 the 𝜉-chromatic mean of a closed Helm graph is

and the 𝜉 -chromatic variance of the closed helm graph

.

The following theorem deals with the 𝜉-chromatic mean and variance of closed helm graph for all other possible values of 𝑛.

Theorem 2.7 For 𝑛 ≥ 7, the 𝜉-chromatic mean of a closed helm graph is

, 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 3) ,

, 𝑖𝑓 𝑛 ≡ 2 (𝑚𝑜𝑑 3) .

and the 𝜉-chromatic variance of the closed helm graph is

, 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 3) ,

, 𝑖𝑓 𝑛 ≡ 2 (𝑚𝑜𝑑 3) .

Proof. The chromatic number of closed helm graph is 𝑛 + 1 that is 𝜉(𝐶𝐻 𝑛 ) = 𝑛 + 1

Case 1: Let 𝑛 ≡ 0(𝑚𝑜𝑑 3). Then the corresponding 𝑝. 𝑚. 𝑓 is

, 𝑖𝑓 𝑖 = 4,5,

, 𝑖𝑓 𝑖 = 7,8,9 … , 𝑛 + 1 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 .

Thus, the 𝜉-chromatic mean is given by 𝜇 𝜉 (𝐶𝐻 𝑛 ) = (1 + 2 + 3)

and the 𝜉 -chromatic variance is given by

.

Case 2: Let 𝑛 ≡ 1(𝑚𝑜𝑑 3). Then, the corresponding 𝑝. 𝑚. 𝑓 is given by

, 𝑖𝑓 𝑖 =

, 𝑖𝑓 𝑖 = 5,6,7 … , 𝑛 + 1 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 .

Thus, the 𝜉 -chromatic mean is given by 𝜇 𝜉 (𝐶𝐻 𝑛 ) = (1 + 2 + 3)

and the 𝜉 -chromatic variance is given by 𝜎 𝜉 2 (𝐶𝐻 𝑛 ) =

(

.

Case 3: Let 𝑛 ≡ 2(𝑚𝑜𝑑 3). The corresponding 𝑝. 𝑚. 𝑓 is given by

, 𝑖𝑓 𝑖 = 4,5 , .

The 𝜉 + -chromatic mean and variance remains same when 𝑛 = 3 since it follows uniform discrete distribution. When 𝑛 = 4,5,6 the 𝜉 + -chromatic mean of a closed helm graph is

and the 𝜉 + -chromatic variance of the closed helm graph is𝜎 𝜉 + 2 (𝐶𝐻 𝑛 ) = 5𝑛 , 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 3) ,

, 𝑖𝑓 𝑛 ≡ 2 (𝑚𝑜𝑑 3) .

and the 𝜉 + -chromatic variance of the closed helm graph is

, 𝑖𝑓 𝑛 ≡ 0(𝑚𝑜𝑑 3) ,

, 𝑖𝑓 𝑛 ≡ 2 (𝑚𝑜𝑑 3) .

Proof. The chromatic number of the closed helm graph is 𝑛 + 1 that is 𝜉 + (𝐶𝐻 𝑛 ) = 𝑛 + 1. Here, we need to consider the following cases.

Case 1: Let 𝑛 ≡ 0(𝑚𝑜𝑑 3). Then, the corresponding 𝑝. 𝑚. 𝑓 is

, 𝑖𝑓 𝑖 = 𝑛 + 1, 𝑛, 𝑛 -1 , Thus, the 𝜉 + -chromatic mean is given by 𝜇 𝜉 + (𝐶𝐻 𝑛 ) = (𝑛 + 1 + 𝑛 + 𝑛 -1)

and the 𝜉 + -chromatic variance is given by 𝜎 𝜉 + 2 (𝐶𝐻 𝑛 ) = (((𝑛 + 1) 2 + (𝑛

)) -

.

Case 2: Let 𝑛 ≡ 1(𝑚𝑜𝑑 3). Then, the corresponding 𝑝. 𝑚. 𝑓 is

, 𝑖𝑓 𝑖 = 𝑛 + 1, 𝑛, 𝑛 -

, 𝑖𝑓 𝑖 = 𝑛 -

, 𝑖𝑓 𝑖 = 𝑛 -3, 𝑛 -4 … ,1 , 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 . Thus, the 𝜉 + -chromatic mean is given by 𝜇 𝜉 + (𝐶𝐻 𝑛 ) = (𝑛 + 1 + 𝑛 + 𝑛 -1) , 𝑖𝑓 𝑖 = 𝑛 + 1, 𝑛, 𝑛 -

, 𝑖𝑓 𝑖 = 𝑛 -2, 𝑛 -

, 𝑖𝑓 𝑖 = 𝑛 -4, 𝑛 -5 … ,

Thus, the 𝜉 + -chromatic mean is given by 𝜇 𝜉 + (𝐶𝐻 𝑛 ) = (𝑛 + 1 + 𝑛 + 𝑛 -1) .

Proof. Note that the chromatic number of flower graph is 2𝑛 + 1 since the central vertex is adjacent to all other vertices. Therefore, all vertices must receive different colors. Thus, the coloring follows uniform discrete distribution. Hence the mean and variance is calculated as 𝑛 + 1 and .

The 𝜉 + -chromatic mean and variance of flower graph is same as 𝜉-chromatic mean and variance of flower graph since it follows uniform discrete distribution.


M R Raksha
email: rakshamr94@gmail.com

P Hithavarshini
email: hithavarshinip@gmail.com

N K Sudev
email: sudev.nk@christuniversity.in4charles.dominic@christuniversity.in

C Dominic

ON 𝒃-COLORING PARAMETERS OF SOME CLASSES OF GRAPHS

Keywords: 𝑏-coloring, coloring mean, coloring variance, 𝑏-chromatic mean, 𝑏chromatic variance. Mathematics Subject Classification: 05C15, 05C75

Vertex coloring has always been a topic of interest. Motivated by the studies on 𝑏-chromatic mean and variance of some standard graphs, in this paper, we obtain few results for 𝑏-chromatic and 𝑏 + -chromatic mean and variance of some cycle related graph classes. Here, Vertex coloring of a graph 𝐺 is taken to be the random experiment. Discrete random variable 𝑋 for this random experiment is the color of randomly chosen vertex of 𝐺.

Introduction

In literature, we go through many problems where the elements (vertices, edges and faces) are assigned with colors, labels or weights subject to certain conditions. This assignment is called graph coloring. In the current discussion, by the term graph coloring, we mean the assignment of colors to the vertices of the graph under consideration. Many coloring problems have been discussed in detail over a couple of centuries.

For the concepts and terminology of graph theory, we refer to [START_REF] West | Introduction to graph theory[END_REF][START_REF] Haray | Graph theory[END_REF]. For further reading on graph coloring, reader are advised to refer to [START_REF] Jensen | Graph coloring problems[END_REF]. Vertices are colored in such a way that no two adjacent vertices gets the same color, this is known as the proper coloring of a graph.

The chromatic number of a graph 𝐺 is the minimum number of colors required to properly color the graph 𝐺. It is denoted by 𝜒(𝐺). A proper 𝑘 -𝑐𝑜𝑙𝑜𝑟𝑖𝑛𝑔 of a graph 𝐺 is a set of k colors denoted by 𝐶 = {𝑐 1 , 𝑐 2 , 𝑐 3 , … , 𝑐 𝑘 } or 𝐶 = {1,2,3, … , 𝑘}. The set 𝐶 𝑖 represents the set of all vertices in 𝐺 having color 𝑐 𝑖 and is known as the color class of the color 𝑐 𝑖 . Cardinality of each color class 𝐶 𝑖 is called the strength of the color 𝑐 𝑖 and is represented by 𝜃(𝑐 𝑖 ).

The graph coloring (vertex coloring) of a graph 𝐺 can be considered as random experiment with a discrete random variable (d.r.v) 𝑋 for the set 𝐶 = {𝑐 1 , 𝑐 2 , 𝑐 3 , … , 𝑐 𝑘 }, a proper 𝑘-coloring of 𝐺 which denotes the color 𝑐 𝑖 of a randomly selected vertex of 𝐺 (see [START_REF] Sudev | On certain coloring parameters of graphs[END_REF]. Therefore, the real-valued function 𝑓(𝑖) given by

𝑓(𝑖) = ( 𝜃(𝑐 𝑖 ) |𝑉(𝐺)| , 𝑖𝑓 𝑖 = 1,2,3, … , 𝑘 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒.
is the 𝑝. 𝑚. 𝑓 (Probability Mass Function) of the d.r.v 𝑋 (see [START_REF] Sudev | On certain coloring parameters of graphs[END_REF]). Therefore, the concept of mean and variance can be extended to coloring mean and coloring variance of a graph 𝐺 for different types of coloring. In this paper, we study coloring mean and coloring variance with respect to a 𝑏coloring of certain classes of graphs, as an extension to [START_REF] Sudev | Some results on the 𝑏-colouring parameters of graphs[END_REF].

Recall the notion of the 𝑏-coloring of a graph, defined in [START_REF] Kok | The 𝑏-chromatic number of certain graphs and digraphs[END_REF], as follows.

Definition 1 [START_REF] Kok | The 𝑏-chromatic number of certain graphs and digraphs[END_REF] The 𝑏-coloring of a graph 𝐺 is a proper vertex coloring 𝑐 in which each color class contains at least one vertex that is adjacent to at least one vertex of all the other color classes.

Definition 2 [START_REF] Kok | The 𝑏-chromatic number of certain graphs and digraphs[END_REF] The 𝑏-chromatic number of a graph 𝐺 is the maximum number of colors that can be used to properly 𝑏-color a graph and is denoted by 𝜑(𝐺).

Definition 3 [START_REF] Sudev | On certain coloring parameters of graphs[END_REF] The coloring mean of a coloring 𝐶 of a given graph 𝐺, denoted by 𝜇 𝐶 (𝐺), is defined to be

𝜇 𝐶 (𝐺) = ∑ 𝑘 𝑖=1 𝑖𝑓(𝑖) = ∑ 𝑘 𝑖=1 𝑖𝜃(𝐶 𝑖 ) ∑ 𝑘 𝑖=1 𝜃(𝐶 𝑖 )
For a positive integer 𝑟, the 𝑟 𝑡ℎ moment of the coloring 𝐶 is denoted by 𝜇 𝐶 𝑟 (𝐺) and is defined as

𝜇 𝐶 𝑟 (𝐺) = ∑ 𝑘 𝑖=1 𝑖 𝑟 𝑓(𝑖)
Definition 4 [START_REF] Sudev | On certain coloring parameters of graphs[END_REF] The coloring variance of a coloring 𝐶 of a given graph 𝐺, denoted by 𝜎 𝐶 2 (𝐺), is defined to be 𝜎 𝐶 2 (𝐺) = 𝜇 𝐶 2 (𝐺) -(𝜇 𝐶 (𝐺)) 2 Definition 5 [START_REF] Sudev | Some results on the 𝑏-colouring parameters of graphs[END_REF] The 𝜑(𝐺)-𝑐ℎ𝑟𝑜𝑚𝑎𝑡𝑖𝑐 𝑚𝑒𝑎𝑛 or 𝑏 -𝑐ℎ𝑟𝑜𝑚𝑎𝑡𝑖𝑐 𝑚𝑒𝑎𝑛 of a graph 𝐺 indicated by 𝜇 𝜑 (𝐺) is a coloring mean of 𝐺 with minimum proper 𝑏 -coloring 𝐶 of 𝐺 and minimum coloring sum 𝜔 𝐺 .

Definition 6 [START_REF] Sudev | Some results on the 𝑏-colouring parameters of graphs[END_REF] The 𝜑 -𝑐ℎ𝑟𝑜𝑚𝑎𝑡𝑖𝑐 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 or 𝑏 -𝑐ℎ𝑟𝑜𝑚𝑎𝑡𝑖𝑐 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 of a graph 𝐺 indicated by 𝜎 𝜑 2 (𝐺) is a coloring variance of 𝐺 in accordance with minimal b-coloring 𝐶 of 𝐺 such that the coloring sum is minimum.

Definition 7 [START_REF] Sudev | Some results on the 𝑏-colouring parameters of graphs[END_REF] The 𝜑 + -chromatic mean or 𝑏 + -𝑐ℎ𝑟𝑜𝑚𝑎𝑡𝑖𝑐 mean of a graph 𝐺 indicated by 𝜇 𝜑 + (𝐺) is a coloring mean of 𝐺 with proper 𝑏-coloring 𝐶 of 𝐺 and maximum coloring sum 𝜔 𝐺 .

Definition 8 [START_REF] Sudev | Some results on the 𝑏-colouring parameters of graphs[END_REF] The 𝜑 + -chromatic variance or 𝑏 + -𝑐ℎ𝑟𝑜𝑚𝑎𝑡𝑖𝑐 variance of a graph 𝐺 indicated by 𝜎 𝜑 + 2 (𝐺) is a coloring variance of 𝐺 in accordance with 𝑏-coloring 𝐶 of 𝐺 such that the coloring sum is maximum.

Motivated by the studies mentioned above, in this paper, we extend the studies on 𝑏-coloring parameters to some fundamental graphs.

Results and Discussions

In this section, we prove some of the statistical parameters like mean and variance for some of the classes of graphs. 𝑏 + -chromatic mean and 𝑏 + -chromatic variance can be obtained by reversing the coloring pattern Of the graph G using the same 𝑏-coloring which was used for 𝑏-chromatic mean and 𝑏-chromatic variance.

Theorem 1

The 𝑏-chromatic mean 𝜇 𝜑 and variance 𝜎 𝜑 2 of prism graph 𝑌 𝑛 is given by

𝜇 𝜑 (𝑌 𝑛 ) = ( 2, 𝑖𝑓 𝑛 = 3 3𝑛-6 2𝑛 + 14, 𝑖𝑓 𝑛 ≥ 4 𝜎 𝜑 2 (𝑌 𝑛 ) = ( 2 3 , 𝑖𝑓 𝑛 = 3 -167𝑛 2 +452𝑛-36 4𝑛 2 -196, 𝑖𝑓 𝑛 ≥ 4
respectively.

Proof. The 𝑏-chromatic number of 𝑌 3 is 3 and 4 for graph with higher order, Since in 𝑌 3 we have two 𝐾 3 we can use each of the three colors 𝑐 1 , 𝑐 2 , 𝑐 3 twice the corresponding p.m.f is f(i)=1/3; for i=1,2,3 and 0;elsewhere.Therefore from the equation we have 𝜇 𝜙 (𝑌 3 )=2 and 𝜎 𝜙 2 (𝑌 3 )=2/3. Now the 𝑏-chromatic number of prism graph with 𝑛 ≥ 4, we need to color two vertices of 𝑐 4 and 𝑐 3 and the color 𝑐 1 and 𝑐 2 is given to 𝑛 -2 Vertices each. So the corresponding p.m.f

𝑓(𝑖) = ( 𝑛-2 2𝑛 , 𝑖𝑓 𝑖 = 1,2
𝜇 𝜙 (𝑌 𝑛 ) = ( 2, 𝑖𝑓 𝑛 = 3 7(𝑛-2) 2𝑛 + 6, 𝑖𝑓 𝑛 ≥ 4 𝜎 𝜙 2 (𝑌 𝑛 ) = ( 2 3 , 𝑖𝑓 𝑛 = 3 -167𝑛 2 +452𝑛-196 4𝑛 2
-36, 𝑖𝑓 𝑛 ≥ 4 respectively.

Proof. The result can be established by reversing the coloring pattern mentioned in Theorem 1.

Theorem 3

The 𝑏-chromatic mean 𝜇 𝜙 and variance 𝜎 𝜙 2 of flower graph 𝐹 𝑛 is given by Proof. The 𝑏-chromatic number of 𝐶𝐻 𝑛 is 4 when n= 3 or 5 and 5 for all the other n. In 𝐶𝐻 3 two vertices each are colored 𝑐 1 , 𝑐 2 , 𝑐 3 and one vertex gets the color 𝑐 4 . Hence the p.m.f is In 𝐶𝐻 𝑛 when n is odd , 𝑖𝑓 𝑛 ≥ 6, 𝑛 -𝑒𝑣𝑒𝑛 respectively.

𝜇 𝜙 (𝐹 𝑛 ) =
𝑓(𝑖) = (
Proof. The result can be established by reversing the coloring pattern mentioned in Theorem 5.

Theorem 7

The 𝑏-chromatic mean 𝜇 𝜙 and variance 𝜎 𝜙 2 of web graph 𝑊 𝑛 is given by , 𝑖𝑓 𝑖 = 4 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 and its corresponding mean and variance is 𝜇 𝜙 (𝑊 3 ) = 2 and 𝜎 𝜙 2 (𝑊 3 ) = 1 respectively.

𝜇 𝜙 (𝑊 𝑛 ) = ( 2,
In 𝑊 5 colors 𝑐 1 is given to 7 vertices, 𝑐 2 is given to 6 vertices, 𝑐 3 is given to two vertices and 𝑐 4 is given to 1 vertex, Hence the p.m.f is In 𝑊 𝑛 when n is odd

𝑓(𝑖) =

ForTheorem 4 Theorem 5

 45 2 , 𝑖𝑓 𝑛 ≥ 7 respectively.Proof. The 𝑏-chromatic number of 𝐹 3 is 4 and for all other 𝐹 𝑛 the 𝑏-chromatic number is 5. It is clear that in 𝐹 3 three vertices are colored 𝑐 1 and two vertices with color 𝑐 2 and each of the remaining vertices with color 𝑐 3 and 𝑐 4 . Hence the p.m.f is 𝑓(and corresponding Mean and Variance is 𝜇 𝜙 (𝐹 3 ) = 2 and 𝜎 𝜙 2 𝐹 4 we can color 2 vertices each with 𝑐 1 , 𝑐 2 , 𝑐 3 and 𝑐 4 and remaining one vertex with 𝑐 5 . Therefore the p.m.f is𝑓(𝑖) = ( 2 9, 𝑖𝑓 𝑖 = 1,2,3,4 1 9 , 𝑖𝑓 𝑖 = 5 0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 and the corresponding Mean and variance is 𝜇 𝜙 (𝐹 4 𝐹 5 we can color 3 vertices each with 𝑐 1 , 𝑐 2 and 2 vertices with color 𝑐 3 and 𝑐 4 and remaining one vertex with 𝑐 5 . Therefore the p.m.f is 𝑓(and the corresponding Mean and variance is 𝜇 𝜙 (𝐹 4 𝐹 6 we can color 6 vertices with 𝑐 1 , 2 vertices each with color 𝑐 2 ,𝑐 3 and 𝑐 4 and remaining one vertex with 𝑐 5 . Therefore the p.m.f is 𝑓(and the corresponding Mean and variance is 𝜇 𝜙 (𝐹 4 𝐹 𝑛 ,n ≥ 7 we can color 𝑛 -1 vertices with color 𝑐 1 , 𝑛 -3 vertices each with color 𝑐 2 , 2 vertices each with 𝑐 3 and 𝑐 4 and remaining one vertex with 𝑐 5 . Therefore the p.m.f is and the corresponding Mean and variance is 𝜇 𝜙 (𝐹 𝑛 ) The 𝑏 + -chromatic mean 𝜇 𝜙 and variance 𝜎 𝜙 2 of flower graph 𝐹 𝑛 is given by 𝜇 𝜙 (𝐹 𝑛 ) The result can be established by reversing the coloring pattern mentioned in Theorem 3. The 𝑏-chromatic mean 𝜇 𝜙 and variance 𝜎 𝜙 2 of closed helm graph 𝐶𝐻 𝑛 is given by 𝜇 𝜙 (𝐶𝐻 𝑛 ) 2 , 𝑖𝑓 𝑛 ≥ 6, 𝑛 -𝑒𝑣𝑒𝑛 respectively.

  𝐶𝐻 4 colors 𝑐 1 , 𝑐 2 , 𝑐 3 , 𝑐 4 are given to two vertices each and 𝑐 5 is given to one vertex. Hence the p.m.f is 𝑓(𝑖) = ( and its corresponding mean and variance is 𝜇 𝜙 (𝐶𝐻 4 𝐶𝐻 5 colors 𝑐 1 , 𝑐 2 are given to 4 vertices each 𝑐 3 is given to two vertices and 𝑐 4 is given to 1 vertex, Hence the p.m.f is and its corresponding mean and variance is 𝜇 𝜙 (𝐶𝐻 5 ) = 2 and 𝜎 𝜙 2

2𝑛- 6 2Theorem 6

 66 vertices are colored 𝑐 1 and 𝑐 2 each, 4 vertices are colored 𝑐 3 , 2 vertices are colored 𝑐 4 and one vertex is colored 𝑐 5 .Hence the p.m.f is 𝑓(and its corresponding mean and variance is 𝜇 𝜙 (𝐶𝐻 𝑛 ) 𝑐 2 , 3 vertices are colored 𝑐 3 , 2 vertices are colored 𝑐 4 and one vertex is colored 𝑐 5 .Hence the p.m.f is 𝑓(The 𝑏 + -chromatic mean 𝜇 𝜙 and variance 𝜎 𝜙 2 of closed helm graph 𝐶𝐻 𝑛 is given by 𝜇 𝜙 (𝐶𝐻 𝑛 )

  The 𝑏-chromatic number of 𝑊 𝑛 is 4 when n= 3 or 5 and 5 for all the other n. In 𝑊 3 , 4 vertices are colored 𝑐 1 color,3 vertices are colored 𝑐 2 2vertices are colored 𝑐 3 and one vertex gets the color 𝑐 4 . Hence the p.m.f is

  and its corresponding mean and variance is 𝜇 𝜙 (𝑊 5

Theorem 8 2 ,Theorem 9 In 𝑆𝑓 4 1 2 2 2vertices, 3

 8294123 𝑐 2 , 4 vertices are colored 𝑐 3 , 2 vertices are colored 𝑐 4 and one vertex is colored 𝑐 5 .Hence the p.m.f is 𝑓(and its corresponding mean and variance is 𝜇 𝜙 (𝑊 𝑛 ) 𝑐 2 , 2 vertices are colored 𝑐 3 , 𝑐 4 each and one vertex is colored 𝑐 5 .Hence the p.m.f is and its corresponding mean and variance is 𝜇 𝜙 (𝑊 𝑛 ) = 9𝑛+24 2(3𝑛+1) and 𝜎 𝜙 2 (𝑊 𝑛 ) = 9𝑛 2 +342𝑛-328 4(3𝑛+1) 2 respectively. The 𝑏 + -chromatic mean 𝜇 𝜙 and variance 𝜎 𝜙 2 of web graph 𝑊 𝑛 is given by 𝜇 𝜙 (𝑊 𝑛 ) 𝑖𝑓 𝑛 ≥ 4, 𝑛 -𝑒𝑣𝑒𝑛 respectively. Proof. The result can be established by reversing the coloring pattern mentioned in Theorem 7. The 𝑏-chromatic mean 𝜇 𝜙 and variance 𝜎 𝜙 2 of sunflower graph 𝑆𝑓 𝑛 is given by𝜇 𝜙 (𝑆𝐹 𝑛 )The 𝑏-chromatic number of 𝑆𝑓 𝑛 is 4 when n= 3,it is 5 when n=4 and 6 for all the other n. In 𝑆𝑓 3 each of the color 𝑐 1 , 𝑐 2 , 𝑐 3 are colored to 2 vertices and one vertex gets the color 𝑐 4 . Hence the p.m.f is and its corresponding mean and variance is 𝜇 𝜙 (𝑆𝑓 3 each of the colors 𝑐 1 , 𝑐 2 , 𝑐 3 , 𝑐 4 is given to 2 vertices and 𝑐 5 is given to 1 vertex, Hence the p.m.f is and its corresponding mean and variance is 𝜇 𝜙 (𝑆𝐹 4 𝑆𝐹 6 each of the colors 𝑐 1 , 𝑐 2 is given to 3 vertices,each of the color 𝑐 3 , 𝑐 4 , 𝑐 5 is given to two vertices and 𝑐 6 is given to 1 vertex, Hence the p.m.f is 𝑓(and its corresponding mean and variance is 𝜇 𝜙 (𝑆𝑓 6 ) = 3 and 𝜎 𝜙 2 𝑆𝑓 𝑛 when n is odd 𝑛 -3 vertices are colored 𝑐 1 , each of the color 𝑐 2 , 𝑐 3 are colored to 𝑛vertices, 2 vertices are colored 𝑐 4 , 2 vertices are colored 𝑐 5 and one vertex is colored 𝑐 6 .Hence the p.m.f is 𝑓(n is even 𝑛 -3 vertices are colored 𝑐 1 , each of the color 𝑐 2 , 𝑐 3 are colored to 𝑛vertices are colored 𝑐 4 , 2 vertices are colored 𝑐 5 each and one vertex is colored 𝑐 6 .Hence the p.m.f is

, 𝑖𝑓 𝑛 > 6, 𝑛 -𝑒𝑣𝑒𝑛 respectively.

Proof. The result can be established by reversing the coloring pattern mentioned in Theorem 1.

Conclusion

In this paper, we have discussed the mean and variance of certain graph classes with respect to two types of their 𝑏-colorings. The study can be extended to many other graph classes, graph operations, graph products and derived graphs such as line graphs, central graphs, total graphs etc. These parameters are interesting not only in the theoretical aspects, but in many applications practical problems such as optimisation, distribution, communication networks, transportation etc. All these facts highlight the wide scope for further research in this area.