Jun Kameda
email: kameda@eps.s.u-tokyo.ac.jp

Hugues Raimbourg

Toshihiro Kogure

Gaku Kimura

Low-grade metamorphism around the down-dip limit of seismogenic subduction zones: Example from an ancient accretionary complex in the Shimanto Belt, Japan

Keywords: accretionary prism, low-grade metamorphism, mica, chlorite, X-ray diffraction, transmission electron microscopy

Reactions involving clay minerals during low-grade metamorphism at the depth of an ancient accretionary complex in the Shimanto Belt, Kyushu, Japan, were studied by integrated transmission electron microscopy-energy dispersive X-ray spectroscopy and X-ray diffraction analyses of the bulk rock and clay fraction. The analyzed metasediment (the Kitagawa unit) contain an incipient sub-horizontal slaty cleavage. Illite crystallinity data and mica b dimensions indicate that the conditions of metamorphic deformation were anchizone-epizone grade and intermediate pressure. Cleavage formation was linked to two reactions involving clay minerals:

(1) the recrystallization of 1M-dominant matrix mica, inherited from the original sedimentary fabric, into thick, defect-free 2M 1 packets along cleavage planes; and (2) the formation of chlorite from 7 Å berthierine. Balanced equations among the clay phases, based on compositional data and their relative abundance, suggest that the decomposition of matrix mica resulted in the formation of paragenetic mica and chlorite along the cleavage planes, without significant elemental outflux. Although a modal increase in phyllosilicates is not indicated by the data, the growth of chlorite and mica along cleavage planes may have a large influence on the rheological properties of a décollement and may be related to the occurrence of the seismic-aseismic transition at ~350 °C.

Introduction

The mechanical properties of sediments entering a subduction zone are progressively modified by dewatering, lithification, and deformation [START_REF] Cowan | Deformation of partly dewatered and consolidated Franciscan sediments near Piederas Blancas Point, California[END_REF][START_REF] Byrne | Structural evolution of mélange terranes in the Ghost Rocks Formation, Kodiak Island, Alaska, in Mélanges, their origin and significance[END_REF][START_REF] Taira | Sediment deformation and hydrology of the Nankai Trough accretionary prism: synthesis of shipboard results of ODP Leg 131[END_REF]Moore et al., 2001;Moore and Saffer, 2001;[START_REF] Hashimoto | Tectonic lithification of sandstone prior to th eonset of seismogenic zone: evidence of tectonic mélange of the Shimanto Belt, Japan. Geochem[END_REF][START_REF] Kimura | Transition of accretionary wedge structures around the up-dip limit of the seismogenic subduction zone[END_REF].

Such modification along a plate boundary fault (i.e., a décollement) has a major influence on several aspects of accretionary prism dynamics, such as wedge geometry, internal structure, and seismicity [START_REF] Wang | Accretionary prism in subduction earthquake cycles: The theory of dynamic Coulomb wedge[END_REF][START_REF] Hyndman | The seismogenic zone of subduction thrust faults: what we know and don't know[END_REF][START_REF] Kimura | Transition of accretionary wedge structures around the up-dip limit of the seismogenic subduction zone[END_REF]. In general, a décollement can be divided into three segments in terms of its mechanical regime: shallow and deep aseismic portions, and an intermediate seismic region [START_REF] Hyndman | The seismogenic zone of subduction thrust faults[END_REF][START_REF] Hyndman | The seismogenic zone of subduction thrust faults: what we know and don't know[END_REF]. The upper of the two seismic-aseismic transitions occurs at temperatures of ~150 °C and is believed to be related to dewatering of subducted sediments due to tectonic loading and diagenetic dehydration reactions (e.g., smectite-illite transition and opal-quartz transition), which allow increase in effective normal stress of the décollement (Moore and Saffer, 2001;[START_REF] Spinelli | Along-strike variations in underthrust sediment dewatering on the Nicoya margin, Costa Rica related to the updip limit of seismicity[END_REF].

Moreover, active carbonate, clay and quartz cementations around this temperature condition may cause sliding instability of the sediments (Moore and Saffer, 2001). On the other hand, onset of the aseismic behavior above ~350 °C is inferred to be related to the mechanical properties of subducted sediments subjected to low-grade metamorphism [START_REF] Hyndman | The seismogenic zone of subduction thrust faults[END_REF][START_REF] Hyndman | The seismogenic zone of subduction thrust faults: what we know and don't know[END_REF].

Although the nature of deep crustal structure in subduction zones remains debated, several seismic reflection surveys (particularly those undertaken at the Cascadia margin) suggest that the décollement widens with increasing depth to be several kilometers thick, forming reflection banding in a seismic profile (e.g., [START_REF] Nedimovic | Reflection signature of seismic and aseismic slip on the northern Cascadia subduction interface[END_REF][START_REF] Calvert | Local thickening of the Cascadia forarc crust and the origin of seismic reflectors in the uppermost mantle[END_REF]. A candidate material for such reflectors is plastically deformed metasediments affected by low-grade metamorphism, with a structure similar to that of mylonite zones. Therefore, phase transitions in subducted sediments, particularly those for phyllosilicates, are likely to influence aseismic faulting along deep décollements [START_REF] Hyndman | The seismogenic zone of subduction thrust faults[END_REF][START_REF] Hyndman | The seismogenic zone of subduction thrust faults: what we know and don't know[END_REF]. Nevertheless, the brittle-ductile transition is rarely exposed at the surface, as most accretionary prism rocks do not reach the transition or are overprinted by late-stage higher-grade metamorphism. Consequently, little is known of the prograde evolution of mineral assemblages and their reaction pathways around the down-dip part of the seismogenic zone, which may control the mechanical transition from seismic to aseismic behavior.

The Kitagawa unit, exposed on the easternmost side of the Kyushu Shimanto Belt, Japan, provides an opportunity to trace the progress of low-grade metamorphism within a subduction zone, because this unit records the highest metamorphic grade of the rocks within the Shimanto Belt (320-330 °C; [START_REF] Kondo | Deformation and fluid flow of a major out-of-sequence thrust located at seismogenic depth in an accretionary complex: Nobeoka Thrust in the Shimanto Belt, Kyushu, Japan[END_REF] and comprises a complete succession ranging from shale/sandstone to intensely cleaved metasediment [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF]. A structural analysis of the unit indicates that vertical loading, probably in a setting around the plate interface, produced the ubiquitous subhorizontal cleavage defined by networks of recrystallized chlorite and mica [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF].

Illite crystallinity (IC) measurements on micas of the Kitagawa unit [START_REF] Mukoyoshi | Quantitative estimation of temperature conditions for illite crystallinity: Comparison to vitrinite reflectance from the Chichibu and Shimanto accretionary complexes, eastern Kyushu, Southwest Japan[END_REF][START_REF] Mukoyoshi | Style of fluid flow and deformation in and around an ancient out-of-sequence thrust: An example from the Nobeoka Teotonic Line in the Shimanto accretionary complex[END_REF] yield lower values at a site located close to the Nobeoka Thrust (0.24 Δ°) compared with a site located more than 5 km from the thrust (0.31 Δ°). Because the intensity of deformation in this unit shows an increase toward the Nobeoka Thrust [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF], the observed trend in IC values is inferred to reflect the extent of mica growth, although additional analyses are needed to characterize the complete sequence of metamorphic processes that occurs during the formation of a mature metapelite.

The aim of the present study is to clarify several aspects of low-grade metamorphism of the Kitagawa unit, including its metamorphic setting, evolution of the mineral assemblage and bulk chemistry, and the reaction pathways of clay minerals. To this end, we subjected bulk samples and the clay fraction (<2 μm) to quantitative analyses by X-ray diffraction (XRD) and high-resolution transmission electron microscopy (HR-TEM), for samples that record varying degrees of deformation. Based on these data, we discuss the influence of low-grade metamorphism on the aseismic behavior of deep décollements.

Geological setting

The Shimanto Belt is an ancient accretionary complex that extends for 800 km along the Pacific side of southwest Japan (Fig. 1a). The belt is divided into northern and southern sub-belts, separated by the Aki Tectonic Line in the Shikoku and Kii regions, and by the Nobeoka Thrust in Kyushu. In eastern Kyushu, the Kitagawa Group was exhumed north of the Nobeoka Thrust, emplaced over the southern Hyuga Group (Fig. 1b). Metamorphic mineral assemblages suggest that the northern belt was metamorphosed in the greenschist facies [START_REF] Toriumi | Tectono-Metamorphism of the Shimanto Belt[END_REF], whereas the southern belt was metamorphosed in the prehnite-pumpellyite to zeolite facies.

The Kitagawa unit, a turbiditic sequence of alternating sandstone-and mudstone-rich layers that contain Eocene radiolarian fossils (Ogawaguchi et al., 1984), records two stages of deformation: an early stage of horizontal contraction, characterized by folding and thrusting, was overprinted by the development of a pervasive slaty cleavage during vertical shortening [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF]. The slaty cleavage is more pronounced in the deeper levels of the Kitagawa unit; i.e., closer to the Nobeoka Thrust.

Vitrinite reflectance values (%Rm) obtained for the Kitagawa unit range from 5.33 to 5.63 [START_REF] Kondo | Deformation and fluid flow of a major out-of-sequence thrust located at seismogenic depth in an accretionary complex: Nobeoka Thrust in the Shimanto Belt, Kyushu, Japan[END_REF], corresponding to maximum paleotemperatures of 320-330 °C based on the method proposed by [START_REF] Sweeney | Evaluation of a simple-model of vitrinite reflectance based on chemical-kinetics[END_REF]. These temperatures are similar to the estimated formation temperature of metamorphic chlorite [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF].

The Hyuga Group, located south of the Nobeoka Thrust, possesses a shale-dominant mélange texture. Vitrinite reflectance data indicate temperatures of 250-270 °C (%Rm = 2.58-2.75; [START_REF] Kondo | Deformation and fluid flow of a major out-of-sequence thrust located at seismogenic depth in an accretionary complex: Nobeoka Thrust in the Shimanto Belt, Kyushu, Japan[END_REF], which are 50-80 °C lower than those obtained for the Kitagawa unit. This difference in temperature suggests that the Nobeoka Thrust was active as an out-of sequence thrust [START_REF] Ohmori | Paleothermal structure of the Shimanto accretionary prism[END_REF], responsible for exhuming the unit from deep within the accretionary complex [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF]. Faulting is thought to have occurred during ~40-48 Ma, as inferred from illite K-Ar and zircon fission track ages [START_REF] Hara | Metamorphic and cooling history of the Shimanto accretionary complex, Kyushu, Southwest Japan: Implication for the timing of out-of-sequence thrusting[END_REF].

Experimental procedure

Samples with various densities of cleavage planes were collected from the Kitagawa unit and were classified into the three groups proposed by [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF]: non-cleaved samples (NM) do not contain slaty cleavage; slightly cleaved samples (SlM) possess both cleavage and bedding; and strongly cleaved samples (StM) contain an intensively developed cleavage with little sign of the original bedding fabric. A total of 13 samples (6 NM samples, 3 SlM samples, and 4 StM samples) were analyzed (Table 1).

Representative portions of each sample were ground using a vibrational mill for 2 minutes under dry conditions for bulk XRD analyses. Each powder, mixed with high-purity corundum (AX-5H, Hinomoto Kenmazai Co., Japan) as an internal standard, was mounted on an XRD glass holder in a side-load manner to minimize any preferred alignment of phyllosilicates. The patterns were recorded using a Rigaku Rint-2000 with monochromatized CuK radiation at 40 kV and 30 mA, with 1° divergence and anti-scattering slits, and a 0.3 mm receiving slit. The 2θ range of 2°-65° was scanned in step scan mode for 4 s, at steps of 0.04° 2θ. The RockJock program, developed by Prof. D.D. Eberl [START_REF] Eberl | User guide to RockJock-A program for determining quantitative mineralogy from X-ray diffraction data[END_REF], was used for quantitative analyses of rock composition. The XRD patterns were fitted using reference patterns for quartz, albite, calcite, pyrite, 1M and 2M 1 micas, and trioctahedral chlorite, as stored in the program.

To obtain the clay size fraction (<2 μm), rock chips were gently crushed by hand and dispersed ultrasonically in distilled water. The clay fraction was then separated by centrifugation and used for oriented analyses of clay minerals in air-dried (AD) and ethylene-glycolated (EG) states. The Newmod program [START_REF] Reynolds | NEWMOD: A computer program for the calculation of one-dimensional diffraction patterns of mixed layered clay minerals: Hanover, New Hampsher[END_REF] was used to interpret the XRD patterns. IC was measured based on the width of the 10 Å peak at half height, under the AD condition [START_REF] Kübler | Evaluation quantitative du métamorphism par la cristallinite de l'illité. Etat des progrèss réalizés ces dernières années[END_REF]. The b dimension of mica was determined using the d-value of the mica 060 peak calibrated by the 221 peak of quartz at 1.541 Å [START_REF] Merriman | Very low-grade metapelites; mineralogy, microtextures and measuring reaction progress[END_REF].

The TEM was operated at 200 kV to observe fine structures in minerals (JEM-2010, JEOL, Japan). Some TEM images were digitized using a CCD camera and processed to remove noisy contrast due to amorphous materials on the specimen surface, using a Wiener-filter [START_REF] Marks | Wiener-filter enhancement of noisy HREM images[END_REF][START_REF] Kilaas | Optical and near-optical filters in high-resolution electron microscopy[END_REF] developed by K. Ishizuka (HREM Research, Inc.), as implemented in Gatan Digital Micrograph version 3.10.0. The clay fraction powders were dispersed on a holey carbon film supported by a Cu-grid for chemical analysis by energy dispersive X-ray spectrometer (EDX) in the TEM (EX-24025JGT, Jeol). To minimize the loss of alkali elements, which is significant in the analysis of clay minerals [START_REF] Van Der Pluijm | Analytical electron microscopy and the problem of potassium diffusion[END_REF], beam diameter was extended to 200 nm and the counting duration was reduced to 30 seconds. Each grain was planarly exposed such that its c* axis was rotated by 20° from the beam axis toward the EDX detector. Quantitative chemical analyses were performed with k-factors for O, Na, Mg, Al, Si, K, Ca, Ti, Mn, and Fe determined from standard samples of biotite, albite, olivine, and hornblende.

The atomic ratio was obtained by correcting for the effect of X-ray absorption by the specimen under the criterion of charge neutrality [START_REF] Van Cappellen | Quantitative transmission X-ray microanalysis of ionic compounds[END_REF], using a program developed by one of the authors (T. Kogure).

Results

Deformation structures within sediment of the Kitagawa unit

Microstructures and deformation stages of the Kitagawa unit have been described in detail by [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF], and are briefly summarized here. The protolith of the low-grade metamorphic rocks is a turbiditic unit that was buried deep within the Shimanto paleo-accretionary prism, down to the base of the inner wedge. The protolith consists of alternating sandstone-and mudstone-rich layers with thicknesses of 2 mm to 10 cm. The sediment consists mainly of detrital quartz and minor plagioclase clasts embedded in a fine-grained matrix. The sandstone-rich layers contain a larger proportion of clasts than do the mudstone-rich layers. A small amount of detrital mica grains is scattered throughout the rock, with the grains being larger in the sandstone-rich layers (Fig. 2a, b, andb'). The microcrystalline matrix is too fine-grained for its mineralogical composition to be determined optically.

Macroscopically, the deformation is characterized by the development of a subhorizontal slaty cleavage that crosscuts and disrupts the original sedimentary fabrics. Structural and magnetic (AMS) analyses indicate that the kinematics of the associated deformation is mainly coaxial shortening, with a compressive axis oriented close to vertical [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF].

At the microscopic scale, the cleavage consists of a dense network of mica and chlorite surrounding quartz and feldspar grains (Fig. 2c andc'). Both of these phyllosilicates have a shape-preferred orientation that defines the foliation, as indicated by their simultaneous extinction when rotating the microscope stage under cross-polarized light (Fig. 2c'). Micas are large (up to 100 μm long), while chlorite grains consist mostly of very thin, entangled fibers.

XRD analyses of the clay fraction

The Kitagawa unit contains a simple and uniform mineral assemblage through the succession, comprising quartz, plagioclase (albite), and phyllosilicates with accessory pyrite, Fe oxides, and rare calcite. However, as shown below, the abundance, composition, and crystallographic features of the phyllosilicates are progressively modified at different stages of cleavage formation. Figure 3 shows typical XRD patterns of the <2 μm size fraction for NM, SlM, and StM samples. The most common phase is dioctahedral mica. Table 1 summarizes the IC values of the analyzed samples. IC values for NM samples show a wide scatter from 0.20 to 0.41 Δ°, while those for SlM samples are 0.22-0.29 Δ°, and those for StM samples are within a limited range around 0.2 Δ°. The lowest of the IC values obtained for NM samples (~0.2 Δ°) probably reflect the presence of detrital mica, whereas a sharpening of the 001 reflection through SlM and StM indicates the thickening of mica grains, which is ascribed mainly to recrystallized mica along cleavage planes. These IC values indicate that the low-grade metamorphism of this unit corresponds to the anchizone-epizone grade (the values that define the boundary between the diagenetic zone and the anchizone, and between the anchizone and epizone, are 0.42 and 0.25 Δ°, respectively; [START_REF] Merriman | Very low-grade metapelites; mineralogy, microtextures and measuring reaction progress[END_REF]. Although the present IC values are not interlaboratory standardized, comparable values have been reported from samples obtained near the Nobeoka Thrust (0.24 Δ°; [START_REF] Mukoyoshi | Style of fluid flow and deformation in and around an ancient out-of-sequence thrust: An example from the Nobeoka Teotonic Line in the Shimanto accretionary complex[END_REF] and from the northern part of the unit (0.31 Δ°; [START_REF] Mukoyoshi | Quantitative estimation of temperature conditions for illite crystallinity: Comparison to vitrinite reflectance from the Chichibu and Shimanto accretionary complexes, eastern Kyushu, Southwest Japan[END_REF], for which the IC indexes were measured using the JIC (Japanese Illite Crystallinity Standard) scale, which was developed for paleothermal analyses of accretionary complexes [START_REF] Hara | New proposal of standard specimens for illite crystallinity measurement: Its usefulness as paleo-geothermal indicator[END_REF]. JIC values are converted to the widely accepted CIS (Crystallinity Index Standard; [START_REF] Warr | Interlaboratory standardization and calibration of clay mineral crystallinity and crystallite size data[END_REF] by the following relation [START_REF] Hara | New proposal of standard specimens for illite crystallinity measurement: Its usefulness as paleo-geothermal indicator[END_REF]: IC (CIS) = 1.55  IC (JIC) -0.07. For instance, the value of 0.24 Δ° (JIC) obtained at the southernmost part of the unit, within intensely cleaved rock [START_REF] Mukoyoshi | Style of fluid flow and deformation in and around an ancient out-of-sequence thrust: An example from the Nobeoka Teotonic Line in the Shimanto accretionary complex[END_REF], is converted to 0.3 Δ° (CIS), corresponding to the metamorphic conditions of the boundary between the low and high anchizone [START_REF] Merriman | Very low-grade metapelites; mineralogy, microtextures and measuring reaction progress[END_REF]. Metapelites of this grade commonly contain an incipient cleavage [START_REF] Merriman | Very low-grade metapelites; mineralogy, microtextures and measuring reaction progress[END_REF], which is consistent with the rocks of the present study area.

The b length of mica has been used as a semi-quantitative barometer of crystallization conditions [START_REF] Sassi | The b 0 value of the potassium white micas as a barometric indicator in low-grade metamorphism of pelitic schists[END_REF][START_REF] Guidotti | Classification and correlation of metamorphic facies series by means of muscovite b 0 data from low-grade metapelites[END_REF]). An increase (decrease) in b length is caused mainly by a greater (smaller) extent of phengite substitution [START_REF] Guidotti | Compositional control on the a and b cell dimensions of 2M 1 muscovite[END_REF]. In the present study, the b values determined from mica 060 reflections range from 9.008 to 9.033 Å (Table 1). The relatively deformed samples yield lower b values (Table 1), suggesting a reduction of phengitic content in recrystallized mica toward more aluminous muscovite.

The relative abundance of different polytypes in micas was qualitatively determined by analyzing random mounts of the clay fraction using XRD (Fig. 4). According to [START_REF] Caillère | Mineralogie des Argiles[END_REF][START_REF] Dalla Torre | Determination of white K-mica polytype ratio-comparison of different XRD methods[END_REF], I(112) of 1M and I(025) of 2M 1 are good indices for estimating the relative abundance of these polytypes because of their low sensitivity to the effect of mica preferred orientation or overlap with the feldspar peak. The intensity relations between the two reflections indicate a progressive transition from 1M to 2M 1 from NM through SlM and StM (Fig. 4).

In addition to mica, chlorite occurs as a clay phase. Although chlorite does not appear in some of the NM samples, the 14 Å and 7 Å peaks become more distinctive with increasing metamorphic grade (Fig. 3). This observation indicates the progressive growth of chlorite within the slaty cleavage. However, the intensity of the 7 Å reflection relative to that of 14 Å is much higher than that of a theoretical chlorite XRD pattern, using a structural formula determined by the present EDX analyses (see Section 4.5.). This discrepancy is probably caused either by peak overlap with another 7 Å phase or the interstratification of 7 Å layers within the chlorite stacking; the latter is more probable, given the broad and diffuse 001 reflection [START_REF] Reynolds | NEWMOD: A computer program for the calculation of one-dimensional diffraction patterns of mixed layered clay minerals: Hanover, New Hampsher[END_REF]. Such a broad 001 peak is more common in NM than in SlM and StM samples (Table 1).

A candidate material for the 7 Å phase in metapelites is kaolin minerals. We tested for the occurrence of kaolin by performing an acid treatment test (2 hours immersed in 1 N HCl at 60 °C). The complete disappearance of the 7 Å peak after the test suggests that the mineral is more likely to be Fe-bearing berthierine, which is highly reactive with acid. The occurrence of chlorite-berthierine, as well as the fine structures of micas, was further examined by TEM (see Section 4.4.).

Quantitative bulk-rock composition determined by RockJock

The nature of prograde changes in the bulk-rock composition was quantitatively analyzed using RockJock [START_REF] Eberl | User guide to RockJock-A program for determining quantitative mineralogy from X-ray diffraction data[END_REF], which can be used to estimate the relative proportions of different polytypes in illite-muscovite [START_REF] Aldega | Detrital illite crystals identified from crystallite thickness measurements in siliciclastic sediments[END_REF]. Accessory minerals (e.g., Fe oxides and rutile) occur in trace amounts (<0.5 wt%) and were excluded from the analyses. Table 1 lists the obtained relative abundances (wt%). Using this program, the total weight percent is a measure of analytical quality, because the content of each mineral is estimated independently with respect to the internal standard of corundum [START_REF] Eberl | User guide to RockJock-A program for determining quantitative mineralogy from X-ray diffraction data[END_REF]. In the present study, the deviation from a total of 100% is within ±2%, except for one sample (Table 1), indicating the high quality of the analyses. The data for the sample that yielded a low total (HN24) were recalculated to 100%, to enable a comparison with the other samples.

Figure 5a shows a clear trend of 1M-to-2M 1 conversion in micas with progressive deformation; NM samples show a wide range of 1M/(1M + 2M 1 ) values, ranging from 0.3 to 1.4, but values are below 0.1 in StM samples. This finding is consistent with XRD analyses of the clay fraction (Fig. 3). This reaction is concomitant with the growth of chlorite (Fig. 5b).

Although chlorite is scarcely distributed and is sometimes absent in NM, it makes up approximately 40% of the mica content in StM samples. Such chlorite growth is also qualitatively compatible with the results of XRD analyses of the <2 μm fraction (Fig. 3).

Variations in the total clay content, which have a strong influence on rheology (e.g., [START_REF] Brown | Compositional and fluid pressure controls on the state of stress on the Nankai subduction thrust: A weak plate boundary[END_REF], are shown as the summation of 1M mica, 2M 1 mica, and chlorite (Fig. 5c). The clay content does not show a large increase with cleavage formation (showing an average change of 1 wt% from NM to StM; Table 1), but the large variation in NM samples is reduced to a uniform composition with cleavage development. The initial variation in NM samples probably reflects variations in grain size (i.e., the ratio of clay-to silt-size grains) within the bulk samples, which are subsequently homogenized in terms of rock composition and fabric by the recrystallization and neocrystallization of phyllosilicates along cleavage planes, thereby diminishing the original sedimentary-related variations.

Microtextural and crystallographic analyses of phyllosilicates by TEM

The analyzed metapelite contains basically two forms of micas [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF]: matrix mica inherited from the original sedimentary fabric, and recrystallized mica that defines the cleavage. TEM images showed that the matrix mica has variable packet thickness, from <30 nm to >100 nm, with several strain features such as voids and layer terminations (white arrows in Fig. 6a). In places, we observed relatively ordered, selected-area electron diffraction (SAED) patterns with dominantly 1M stacking, although accompanied by packets showing two-layer (2M 1 ) periodicities (Fig. 6b). Stacking disorders are visible, even in crystals that yield ordered ED patterns (Fig. 6b).

Recrystallized mica that occurs along cleavage planes has a relatively homogeneous thickness (mostly >100 nm) and has fewer defects than the matrix mica (Fig. 6c). Although 1M was occasionally present, 2M 1 was more commonly observed.

Chlorite occurs as well-defined packets (>100 nm thick) and is oriented parallel to the cleavage (Fig. 7a). Lattice fringe images reveal the presence of 7 Å berthierine (Fig. 7b andc), which is generally intergrown with 14 Å chlorite layers at the monolayer level rather than forming discrete packets. In some cases, berthierine constitutes the larger fraction of a single packet (Fig. 7b). The interstratification of berthierine and chlorite is commonly observed in diagenetic to low-grade metamorphic rocks [START_REF] Lee | Chlorite-illite/muscovite interlayered and interstratified crystals: A TEM/STEM study[END_REF][START_REF] Ahn | Transmission electron microscopic study of diagenetic chlorite in Gulf Coast argillaceous sediments[END_REF][START_REF] López-Munguira | Transmission electron microscopy study of very low-grade metamorphic rocks in Cambrian sandstones and shales, Ossa-Morena zone, southwest Spain[END_REF] and in synthetic samples [START_REF] Velde | Phase equilibria studies in the system MgO-Al 2 O 3 -SiO 2 -H 2 O: chlorite and associated minerals[END_REF][START_REF] James | Stability and phase relations of iron chlorite below 8.5 kb P H2O[END_REF].

The interstratified phases are considered to represent a transient state toward a mature chlorite. In Fig. 7d, two berthierine layers (TOTO) are transformed to a single chlorite layer (TOT and one brucite-like B layer) by tetrahedral inversion (white bracket in the figure), strongly suggesting a solid-state conversion, as discussed in detail by [START_REF] Xu | Interstratification and other reaction microstructures in the chlorite-berthierine series[END_REF].

TEM-EDX chemical analyses

The chemical compositions of chlorite in the present samples were measured by [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF] using electron microprobe analyzer (EMPA), and were employed to estimate the formation temperature of chlorite based on the chemical thermometer developed by [START_REF] Vidal | Exhumation paths of high-pressure metapelites obtained from local equiribria for chlorite-phengite assemblages[END_REF]. However, the grain size of clays is generally too small to be measured satisfactorily by EMPA; thus, we re-examined the composition of mica and chlorite by EDX in TEM (Tables 2 and3). The Si content of micas in NM samples ranges from 3.0 to 3.5 if normalized to O 10 (OH) 2 , but the range of values is smaller in SlM and StM samples (3.1-3.3), which have a high interlayer charge; i.e., Na + K = ~1.0 (Fig. 8a). The interlayer composition is characterized by a low Na content (Na/(Na + K) < 0.15) for all deformation stages (Table 2). In addition to a low interlayer deficiency, SlM and StM samples have low Mg + Fe contents (Fig. 8b) and a small Fe/(Fe + Mg) ratio (Fig. 8c) compared with NM. The depletion in mafic elements is consistent with the low phengitic content inferred from the lower b dimensions in SlM and StM samples (Table 1).

Figure 9 shows the chemical compositions of chlorite determined by TEM-EDX, along with data measured by EMPA [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF]. The two techniques yield comparable results, indicating that element contamination during EMPA is not significant. There is almost no compositional variation among the samples of different metamorphic grade.

Discussion

Clay mineral reactions in the Kitagawa unit during anchizone-epizone grade metamorphism

The exposure of a series of lithologies ranging from shale/sandstone with well-preserved sedimentary fabrics to moderately or intensely deformed slates enables a discussion on the change in mineralogy that occurs during cleavage formation in the Kitagawa unit. As shown above, clay minerals in the unit are restricted to two types: dioctahedral micas and trioctahedral chlorite. Although many factors (e.g., temperature, pressure, bulk rock composition, and fluid-rock ratio) may affect the mineral assemblage and its evolution in low-grade metamorphic rocks [START_REF] Merriman | Very low-grade metapelites; mineralogy, microtextures and measuring reaction progress[END_REF], [START_REF] Merriman | Contrasting clay mineral assemblages in British Lower Paleozoic slate belts: the influence of geotectonic setting[END_REF] showed that the clay mineralogy of a slate belt in the British Isles is strongly dependent on the regional geotectonic setting. Many different species of clay minerals can be observed in extensional basins, including the aluminous kaolin-pyropyllite and sodium mica-paragonite series, as well as dioctahedral K-mica and the trioctahedral 2:1 series. This variation in clay mineralogy is due to intense seawater circulation or the intrusion of volcanic rocks, especially during sedimentation [START_REF] Merriman | Contrasting clay mineral assemblages in British Lower Paleozoic slate belts: the influence of geotectonic setting[END_REF]. In contrast, the restricted circulation of seawater or absence of high-Na/K fluids may reduce such diversity in convergent basins [START_REF] Merriman | Contrasting clay mineral assemblages in British Lower Paleozoic slate belts: the influence of geotectonic setting[END_REF]. This may explain the poor diversity of clay minerals in the present samples, as well as the absence of kaolin and pyrophyllite, and the low sodium content in the mica interlayer.

Despite the invariant K-dominant interlayer composition, the crystallographic features of mica are modified by deformation-induced recrystallization. Bulk-rock XRD analyses and microscopic observations by TEM suggest that the matrix mica, which is 1M, is converted to 2M 1 packets along cleavage planes. In addition, strain-related structures are healed to form thick, defect-free packets, which is also indicated by the low IC values obtained for deformed samples.

Such crystallographic maturation has been widely recognized in anchizone-grade metapelites that show the development of incipient slaty cleavage [START_REF] Lee | Evidence for syntectonic crystallization for the mudstone to slate transition at Leigh Gap, Pennsylvania, USA[END_REF][START_REF] Li | TEM and AEM constraints on the origin and significance of chlorite-mica stacks in slates: an example from Central Wales[END_REF].

Unlike the characteristic conversion of micas, the origin of chlorite is somewhat problematic.

Chlorite in metapelites is commonly derived from a prograde reaction that originates from trioctahedral smectite via corrensite (e.g., Hillier, 1993;[START_REF] Schmidt | HRTEM and SAED investigations of polytypism, stacking disorder, crystal growth, and vacancies in chlorites from subgreenschist facies outcrops[END_REF]. However, the formation of Fe-Mg-rich smectite within an accretionary prism is generally localized to specific lithologies such as incorporated oceanic crust (e.g., Kameda et al., in review), which is not found in the Kitagawa unit. Previous studies have noted that chlorite forms via the conversion of dioctahedral smectite to illite [START_REF] Hower | Mechanism of burial metamorphism of argillaceous sediments: 1. Mineralogical and chemical evidence[END_REF][START_REF] Ahn | Transmission electron microscopic study of diagenetic chlorite in Gulf Coast argillaceous sediments[END_REF], because of the typical depletion of mafic elements in illite relative to the starting smectite [START_REF] Drief | Chemical composition of smectites formed in clastic sediments. Implication for the smectite-illite transformation[END_REF]. In the present case, however, the chlorite content is generally low prior to deformation (Fig. 5b) and it is locally absent, despite the completion of smectite-to-illite conversion, which typically occurs at <150 °C. This reaction may have contributed to some of the chlorite in the NM samples, but the progressive growth of chlorite from the SlM to StM samples (Fig. 11b) requires another process, possibly the decomposition of matrix mica of phengitic composition to form paragenetic mica and chlorite along cleavage planes. The likelihood of this reaction is evaluated with reference to the following balanced equation. We roughly consider that the increment of chlorite from NM to StM samples (by 10 wt% on average) and the total mica content in StM samples ( 22wt%) reflect the effects of recrystallization. Based on the chemical composition determined by EDX (Tables 2 and3), the molar weight of mica and chlorite per structural formula in StM samples is 395 and 636, respectively. Then, the molar ratio of the final product between the total mica content in StM samples (22 wt%) versus the increase in chlorite content from NM to StM samples (10 wt%) is estimated to be 22/395:10/636 = 1:0.275. Assuming the amount of Al 2 O 3 to be constant throughout the reaction, the molar ratio of the starting material (mica in NM samples) required to complete the reaction is derived from the flowing equation: (1).

This equation indicates that about 30 wt% (= 22 wt%  1.35) of the starting material is needed, which is compatible with the average mica content in NM samples (31 wt%). The mass balance results, based on Al conservation, support the validity of reaction (1). However, this reaction requires the presence of Mg 2+ + Fe 2+ . The origin of these elements is uncertain, but may have been other mafic minerals such as detrital biotite, pyrite, or Fe-oxides.

When this reaction proceeds, berthierine may occur as a precursor. As mentioned above, the XRD patterns of the oriented clay preparations show a weak, broad 14 Å reflection relative to pure chlorite (001), which is ascribed to interstratification between berthierine and chlorite.

Hillier and Velde (1992) used NEWMOD [START_REF] Reynolds | NEWMOD: A computer program for the calculation of one-dimensional diffraction patterns of mixed layered clay minerals: Hanover, New Hampsher[END_REF] to calculate the one-dimensional XRD pattern of interstratified chlorite-berthierine with a variable mixture ratio, and obtained a near-linear correlation between berthierine content (5%-35%) and width (°2θ) at half the height of the 14 Å peak, independent of chemistry and crystallite thickness. Our experimental patterns show a sharpening of the 14 Å peak from NM samples (0.25-0.37 of width at half peak height; average, 0.282) to SlM (0.265) and StM (0.192) (Table 1). Based on fig. 7 in [START_REF] Hillier | Chlorite interstratified with a 7 a mineral: An example from offshore Norway and possible implications for the interpretation of the composition of diagenetic chlorites[END_REF], NM is estimated to contain >5% berthierine. The reduction in berthierine content is interpreted to reflect post-precipitated conversion from berthierine layers to mature chlorite in a solid-state manner, as observed by TEM (Fig. 7d).

Element flux during cleavage formation

The mass flux of elements during slate formation has been extensively studied in recent decades. For example, studies of kinematic strain (e.g., Write and Platt, 1982;Beutner and Charles, 1985;[START_REF] Brandon | Volume strain associated with pressure-solution deformation in sandstones from high P-low T terrains: A third of the rock is missing![END_REF] have shown a significant volume loss (30%-60%) in slates, but [START_REF] Erslev | Limited, localized nonvolatile element flux and volume change in Appalachian slates[END_REF] noted that such estimates are based on a misinterpretation of complicated, heterogeneous strain and metamorphic reactions, and emphasized the near-isochemical nature of the slates, as also indicated by geochemical signatures [START_REF] Shaw | Trace elements in pelitic rocks. Part 1: Variations during metamorphism[END_REF][START_REF] Ague | Evidence for major mass transfer and volume strain during regional metamorphism of pelites[END_REF][START_REF] Wintsch | Open-system, constant-volume development of slaty cleavage, and strain induced replacement reactions in the Martinsburg Formation, Leigh Gap, Pennsylvania[END_REF]. The extent of element flux may be recorded in the resultant mineral assemblage, but few studies have provided quantitative data on slate composition. Ward and Gomez-Fernandez (2003) applied the Rietveld method to Spanish roofing slates, and assessed the applicability of quantitative analyses with respect to the bulk chemical composition, although the authors did not quantify the degree of modal change during slate formation. [START_REF] Lee | Evidence for syntectonic crystallization for the mudstone to slate transition at Leigh Gap, Pennsylvania, USA[END_REF] performed a semi-quantitative XRD analysis of mudstone/slate samples and compared the relative intensity of the constituent minerals, indicating an approximately homogeneous rock composition across the lithologic trnsition. The quantitative data of the present study indicate a near-constant modal composition, considering the average values for samples from each deformation stage (Table 1). For example, the content of quartz, which is one of the main mobile elements, deviates by just ~4 wt% from 40 wt%; thus, it is unlikely that volume outflux occurred on the order of several tens of weight percent.

Van de [START_REF] Van De Kamp | Smectite-illite-muscovite transformations, quartz dissolution, and silica release in shales[END_REF] suggested that measurable amounts of SiO 2 (~20 wt%) are released by the illite-to-muscovite conversion. Equation (1) indicates that the production of 1.0 mole of cleavage-parallel mica leads to 0.26 moles of SiO 2 release, corresponding to 1 wt% at most (= 22 wt%  0.26  60/395). Thus, in the present case, negligible amounts of SiO 2 were released via reactions involving clay minerals. [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF] performed XRF analyses of bulk samples (the same samples as those analyzed in the present study), revealing that the concentration of immobile TiO 2 is inversely related to the concentration of mobile SiO 2 . The authors proposed that any outflow of SiO 2 would have been caused primarily by quartz dissolution. However, in terms of the relation between SiO 2 content and deformation, their data show a high degree of scatter. This result, combined with the constant modal composition of samples of variable metamorphic grade, suggests that SiO 2 outflow occurred prior to cleavage formation, during diagenesis at shallow levels (e.g., [START_REF] Renard | An integrated model for transitional pressure solution in sandstones[END_REF][START_REF] Yang | Pressure solution in sedimentary basins: effect of temperature gradient[END_REF]Kawabata et al., 2007); thus, high fluxes of elements would not have occurred during anchizone metamorphism.

Geotectonic setting of the Kitagawa unit and implications for the weak and aseismic nature of the deep décollement

Previous studies have estimated the peak paleotemperature of the Kitagawa unit. Based on vitrinite reflectance values, [START_REF] Kondo | Deformation and fluid flow of a major out-of-sequence thrust located at seismogenic depth in an accretionary complex: Nobeoka Thrust in the Shimanto Belt, Kyushu, Japan[END_REF] estimated thermal maturation at 320-330 °C.

Application of the chlorite geothermometer developed by [START_REF] Vidal | Exhumation paths of high-pressure metapelites obtained from local equiribria for chlorite-phengite assemblages[END_REF] yielded an average temperature of ~272 °C [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF], although the data show a wide scatter corresponding to temperatures of 200-400 °C. [START_REF] Mukoyoshi | Quantitative estimation of temperature conditions for illite crystallinity: Comparison to vitrinite reflectance from the Chichibu and Shimanto accretionary complexes, eastern Kyushu, Southwest Japan[END_REF] and [START_REF] Underwood | A comparison among organic and inorganic indicators of diagenesis and low-temperature metamorphism, Tertiary Shimanto Belt, Shikoku, Japan[END_REF] derived regression equations for IC-temperature relations applicable to paleothermal analysis of the Shimanto accretionary complex. The IC value of 0.24 Δ° obtained near the Nobeoka Thrust [START_REF] Mukoyoshi | Style of fluid flow and deformation in and around an ancient out-of-sequence thrust: An example from the Nobeoka Teotonic Line in the Shimanto accretionary complex[END_REF] corresponds to temperatures of 304±30 °C and 330 °C, respectively. The use of our IC value for StM samples (0.2 Δ°) yields slightly higher temperatures of 312±30 °C and 347 °C. Considering these estimates, a metamorphic temperature of 300-350 °C is an appropriate estimate for the Kitagawa unit.

Although there are few direct constraints on the peak burial depth, [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF] estimated a minimum burial depth of ~6 km based on a high geothermal gradient of 50 °C/km in the Eocene, influenced by the subduction of a young, hot oceanic plate [START_REF] Sakaguchi | Thermal structure and paleo-heat flow in the Shimanto accretionary prism[END_REF]. This high regional geotherm may be overestimated.

The b dimensions of micas in the analyzed samples show a narrow range of 9.01-9.03 Å (9.02 Å on average; Table 1), and StM samples yield values on the lower side of this range (~9.01 Å). According to the regression relation between cell parameters and the composition of 2M 1 mica [START_REF] Guidotti | Compositional control on the a and b cell dimensions of 2M 1 muscovite[END_REF], a b dimension of ~9.01 Å is estimated to correspond to a Mg + Fe content of 0.4 in the case of an interlayer composition of Na/(Na + K) < 0.15, which is consistent with the mica composition determined independently by EDX (Table 2). The consistency of the two parameters indicates that the b dimension is a reliable measure of metamorphic conditions, as suggested by [START_REF] Abad | The structure and the phyllosilicates (chemistry, crystallinity and texture) of Talas Ala-Tau (Tien Shan, Kyrgyz Republic): comparison with more recent subduction complexes[END_REF]. The b dimensions obtained in the present study, which range from 9.01 to 9.03 (Table 1), are within the values of 9.0 and 9.04 assigned by [START_REF] Guidotti | Classification and correlation of metamorphic facies series by means of muscovite b 0 data from low-grade metapelites[END_REF] to the lower/upper limit of the intermediate-pressure facies series for a geothermal gradient of 25-35 °C/km. Although the thermal structure of a subduction zone may show some longitudinal variations rather than being isothermal (e.g., Nankai, [START_REF] Watanabe | Heat flow in the Philippine Sea[END_REF][START_REF] Yamano | Nankai Tourgh: A hot trench?[END_REF][START_REF] Yamano | Extremely high heat flow anomaly in the middle part of the Nankai Trough[END_REF], the Kitagawa unit may have experienced burial in an area with a geothermal gradient of 25-35 °C/km, at least at the time of mica formation. At this time, the minimum burial depth of the Kitagawa unit would have been at least 8.5 km, further supporting the proposal by [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF] that the rock has been transported far below the frontal contractional setting of the prism (4-5 km thick in the case of modern complexes), where it was subjected to vertical compaction [START_REF] Raimbourg | Horizontal shortening versus vertical loading in accretionary prisms[END_REF].

It is uncertain how closely the Kitagawa unit approached the ancient décollement, but temperatures of 300 °C or more suggest a vertical separation of less than 2 km (for a geothermal gradient of 25-30 °C/km), if the unit was located just above the area where the 350 °C isotherm intersects the décollement. This distance (between the Kitagawa unit and the décollement) increases (decreases) as the location of the unit is shifted landward (seaward).

Seismic reflection surveys of the Cascadia margin [START_REF] Nedimovic | Reflection signature of seismic and aseismic slip on the northern Cascadia subduction interface[END_REF][START_REF] Calvert | Local thickening of the Cascadia forarc crust and the origin of seismic reflectors in the uppermost mantle[END_REF] suggest that a deep aseismic décollement has a banded reflection structure, extending regionally at depths of 5-8 km from the top of the subducting slab. These studies proposed that the structure corresponding to this reflector is a ductile shear zone composed of foliated metasediment. It is likely that subhorizontal slaty cleavage within the Kitagawa unit would produce such a seismic structure.

Previous sliding experiments on crustal rock (granite) indicate that the seismic-aseismic transition occurs at ~350 °C, possibly related to pressure solution creep of quartz [START_REF] Blanpied | Fault stability inferred from granite sliding experiments at hydrothermal conditions[END_REF][START_REF] Blanpied | Frictional slip of granite at hydrothermal conditions[END_REF]. This observation may explain the down-dip limit of seismogenic subduction zones [START_REF] Hyndman | The seismogenic zone of subduction thrust faults[END_REF]. However, the sediments analyzed in the present study are rich in phyllosilicates, meaning that it is uncertain whether the rheology of the subducted sediments is controlled by quartz deformation. In the case of the Kitagawa unit, we consider that chlorite growth would have had a marked effect on the bulk-rock rheology, because of the ready activation of (001) glide, giving rise to plastic deformation even below 300 °C [START_REF] Bons | Deformation of chlorite in naturally deformed low-grade rocks[END_REF]. The stable (i.e., aseismic) and low-friction behavior of chlorite [START_REF] Ikari | Frictional and hydrologic properties of clay-rich fault gouge[END_REF]) may be related to such intracrystalline deformation. Although the chlorite content in the present study is <12 wt% within intensely cleaved samples, [START_REF] Collettini | Fault zone fabric and fault weakness[END_REF] experimentally showed that even a small amount of phyllosilicates (in their case, 5% talc) may control the mechanical behavior of a fault zone if they form a dense network, thereby maintaining the long-term shear strength at a reasonably low level. In the present case, the occurrence of a well-organized network of micas and chlorite may explain the aseismic behavior of the deep décollement.

Conclusion

Our mineralogical study on the Kitagawa unit of the Shimanto accretionary prism showed that the growth of paragenetic mica and chlorite during anchizonal-epizonal low-grade metamorphism contributed to the development of a pervasive slaty cleavage. The reactions proceeded without large elemental flux or modal fluctuations in phyllosilicate content. The mechanical transition from seismic to aseismic behavior along the deep décollement is attributable in part to the progressive development of subhorizontal mica and chlorite networks, similar to those observed at the surface. [START_REF] Eberl | User guide to RockJock-A program for determining quantitative mineralogy from X-ray diffraction data[END_REF].

Figure captions

  1.35  (K 0.83 Na 0.06 )(Fe 0.17 Mg 0.28 Al 1.64 )(Si 3.22 Al 0.70 Ti 0.08 )O 10 (OH) 2  (K 0.92 Na 0.05 )(Fe 0.1 Mg 0.26 Al 1.68 )(Si 3.24 Al 0.72 Ti 0.04 )O 10 (OH) 2 + 0.275  (Al 1.54 Fe 2.7 Mg 1.6 )(Si 2.70 Al 1.22 Ti 0.07 )O 10 (OH) 8 + 0.16K + + 0.023Na + -0.45Fe 2+ -0.24Mg 2+ + 0.03Ti 4+ + 0.27Si 4+

Fig. 1 .

 1 Fig. 1. Geological map of the Kitagawa unit in the northern Shimanto Belt, Kyushu, Japan. The

Fig. 2 .

 2 Fig. 2. Microstructural evolution from the incipient development of a horizontal foliation (a, b,

Fig. 3 .

 3 Fig. 3. Typical XRD patterns for oriented preparations of the <2 μm fraction in the EG state.

Fig. 5 .

 5 Fig. 5. Evolution of the 1M-to-2M 1 ratio of micas (1M/(1M + 2M 1 )) (a), chlorite/mica (b), and

Fig. 6 .

 6 Fig. 6. (a) TEM image of a mica packet in an NM sample containing pervasive defect structures.

Fig. 7 .

 7 Fig. 7. (a) Thick packet of chlorite in an StM sample. (b) Lattice fringe image of a packet

Fig. 8 .

 8 Fig. 8. Chemical compositions of micas plotted as a function of Si. (a) Interlayer cations, total

Table 1 .

 1 Crystal-chemical parameters and quantitative rock compositions determined by the program RockJock

Table 2 .

 2 Chemical compositions of micas normalized to O 10 (OH) 2

Table 3 .

 3 Chemical compositions of chlorite normalized to O 10 (OH) 8

Table 2 .

 2

			NM				SlM			StM
		14_04 HN18 HN24	avg.	16_05 HN25	avg.	HN34 HN40	avg.
	n	33	17	20	70	17	16	33	17	7	25
	Si	3.20	3.26	3.23	3.22	3.21	3.28	3.24	3.25	3.21	3.24
	IV Al	1.66	1.60	1.64	1.64	1.49	1.63	1.56	1.66	1.72	1.68
	VI Al	0.72	0.71	0.66	0.70	0.76	0.67	0.72	0.71	0.74	0.72
	Fe	0.14	0.20	0.19	0.17	0.13	0.14	0.13	0.11	0.07	0.10
	Mg	0.29	0.29	0.25	0.28	0.29	0.28	0.28	0.27	0.24	0.26
	Ti	0.08	0.03	0.11	0.08	0.03	0.05	0.04	0.04	0.06	0.04
	K	0.83	0.89	0.83	0.84	0.91	0.92	0.91	0.92	0.91	0.92
	Na	0.07	0.05	0.42	0.06	0.05	0.05	0.05	0.05	0.04	0.05
	∑inter	0.90	0.94	0.87	0.90	0.96	0.97	0.96	0.98	0.95	0.97
		SlM			StM
		16_05	16_07 HN34 avg.
	n	5	17		7	24
	Si	2.84	2.73	2.68	2.71
	IV Al	1.53	1.53	1.55	1.54
	VI Al	1.12	1.20	1.26	1.22
	Fe	2.67	2.69	2.71	2.70
	Mg	1.59	1.61	1.60	1.60
	Ti	0.03	0.07	0.06	0.07

Table 3 .

 3

Acknowledgements

Fabrizio Storti and two anonymous reviewers are thanked for their constructive comments which helped to improve the manuscript. This work was financially supported by a Grant-in-Aid for Scientific Research from MEXT-KAKENHI 21107005.


Aline Castro

Sandi Klavžar
email: sandi.klavzar@fmf.uni-lj.si

Michel Mollard
email: michel.mollard@ujf-grenoble.fr

Yoomi Rho
email: rho@incheon.ac.kr

On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes

Keywords: Fibonacci cubes, Lucas cubes, domination number, 2-packing number, automorphism group Math. Subj. Class. (2010): 05C69, 05C25

Let Γ n and Λ n be the n-dimensional Fibonacci cube and Lucas cube, respectively. The domination number γ of Fibonacci cubes and Lucas cubes is studied. In particular it is proved that γ(Λ n ) is bounded below by Ln-2n n-3

, where L n is the n-th Lucas number. The 2-packing number ρ of these cubes is also studied. It is proved that

and the exact values of ρ(Γ n ) and ρ(Λ n ) are obtained for n ≤ 10. It is also shown that Aut(Γ n ) Z 2 .

Introduction

Fibonacci cubes form a class of graphs introduced because of their properties applicable for interconnection networks [START_REF] Hsu | Fibonacci cubes-a new interconnection technology[END_REF]. Lucas cubes [START_REF] Munarini | On the Lucas cubes[END_REF] are subgraphs of Fibonacci cubes in which certain "non-symmetric" vertices are removed. In this way we get graphs with more symmetries, a fact that will be further justified in this paper. Both classes of cubes have been considered from various points of view, see [START_REF] Dedó | The observability of the Fibonacci and the Lucas cubes[END_REF][START_REF] Ellis-Monaghan | Decycling of Fibonacci cubes[END_REF][START_REF] Gregor | Recursive fault-tolerance of Fibonacci cube in hypercubes[END_REF][START_REF] Klavžar | Fibonacci cubes are the resonance graphs of Fibonaccenes[END_REF][START_REF] Munarini | Structural and enumerative properties of the Fibonacci cubes[END_REF][START_REF] Taranenko | Fast recognition of Fibonacci cubes[END_REF].

In this paper we study Fibonacci cubes and Lucas cubes from the viewpoint of domination and packing. While searching for (vertex) subsets of a graph (like dominating sets) it is useful to know symmetries of the graph, hence we first describe automorphism groups of these graphs in Section 2.

In Section 3 we study the domination number of Fibonacci cubes as initiated in [START_REF] Pike | The domination number of Fibonacci cubes[END_REF], and also investigate that of Lucas cubes. We first give some connections between the domination number of Fibonacci cubes and Lucas cubes and construct dominating sets for 9-dimensional cubes. Then we obtain a lower bound on the domination number of Lucas cubes.

A graph invariant closely related to the domination number is the 2-packing number, which is the topic of Section 4. We first obtain an exponential (in terms of the dimension) lower bound on the 2-packing number of the Lucas cubes which is a natural lower bound for the Fibonacci cubes. Combining computer search with some arguments the exact values for the 2-packing number of both classes of cubes up to and including dimension 10 are obtained.

In the rest of this section we define the concepts needed in this paper. For a connected graph G, the distance d G (u, v) (or d(u, v) for short) between vertices u and v is the usual shortest path distance.

Let n ≥ 1 and Q n be the n-dimensional hypercube. A Fibonacci string of length n is a binary string b 1 b 2 . . . b n with b i • b i+1 = 0 for 1 ≤ i < n. In other words, Fibonacci strings are binary strings that contain no consecutive 1's. The Fibonacci cube Γ n , for n ≥ 1 is the subgraph of Q n induced by the Fibonacci strings of length n.

A Fibonacci string b 1 b 2 . . . b n is a Lucas string if b 1 • b n = 0. The Lucas cube Λ n , for n ≥ 1 is the subgraph of Q n induced by the Lucas strings of length n.
It is well-known (cf. [START_REF] Hsu | Fibonacci cubes-a new interconnection technology[END_REF]) that |V (Γ n )| = F n+2 , where F n are the Fibonacci numbers: [START_REF] Munarini | On the Lucas cubes[END_REF], where L n are the Lucas numbers:

F 0 = 0, F 1 = 1, F n = F n-1 + F n-2 for n ≥ 2. Similarly, |V (Λ n )| = L n for n ≥ 1, see
L 0 = 2, L 1 = 1, L n = L n-1 + L n-2 for n ≥ 2.
For n ≥ 1 and 0

≤ k ≤ n, let Γ n,k be the set of vertices of Γ n that contain k 1's. Hence Γ n,k is the set of vertices of Γ n at distance k from 0 n . Λ n,k is defined analogously. In particular, Γ n,0 = Λ n,0 = {0 n } and Γ n,1 = Λ n,1 = {10 n-1 , 010 n-2 , . . . , 0 n-1 1}. If uv ∈ E(Γ n ), where u ∈ Γ n,k and v ∈ Γ n,k-1 (k ≥ 1)
, then we say that v is a down-neighbor of u and that u is an up-neighbor of v. The same terminology again applies to Lucas cubes. Let G be a graph. Then D ⊆ V (G) is a dominating set if every vertex from V (G) \ D is adjacent to some vertex from D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A set X ⊆ V (G) is called a 2-packing if d(u, v) > 2 for any different vertices u and v of X. The 2-packing number ρ(G) is the maximum cardinality of a 2-packing of G. It is well known that for any graph G, γ(G) ≥ ρ(G), cf. [START_REF] Imrich | Topics in Graph Theory[END_REF].

Finally, the automorphism group of a graph G is denoted by Aut(G). For instance, Aut(C n ) = D 2n , where C n is the n-cycle and D 2n is the dihedral group on n elements. Recall that D 2n can be represented as x, y | x 2 = 1, y n = 1, (xy) 2 = 1 .

Automorphism groups

In this section we determine the automorphism groups of Fibonacci cubes and Lucas cubes.

Let n ≥ 1 and define the reverse map r : Γ n → Γ n with:

r(b 1 b 2 . . . b n ) = b R = b n b n-1 . . . b 1 . (1)
It is easy to observe that r is an automorphism of Γ n . We are going to prove that r is the only nontrivial automorphism of Γ n . For this sake, the following lemma is useful.

Lemma 2.1 Let n ≥ 3 and k ≥ 2. Then any different u, v ∈ Γ n,k have different sets of down-neighbors. Proof. Since u, v ∈ Γ n,k , d(u, v) ≥ 2.
We distinguish two cases. Suppose first d(u, v) = 2 and let u and v differ in positions i and j. Since u, v ∈ Γ n,k , we may assume without loss of generality that u i = v j = 1 and u j = v i = 0. Moreover, u and v agree in all the other positions. Since k ≥ 2, there exists an index = i, j such that u = v = 1. Then u + e is a down-neighbor of u but not a down-neighbor of v.

Assume now d(u, v) ≥ 3. Let i be an arbitrary index such that u i = v i . We may assume that u i = 1. Then u + e i is a down-neighbor of u but not of v.

Theorem 2.2 For any n ≥ 1, Aut(Γ n ) Z 2 .
Proof. The assertion is clear for n ≤ 2, hence assume in the rest that n

≥ 3. Let α ∈ Aut(Γ n ). Since 0 n is the only vertex of degree n, α(0 n ) = 0 n . Therefore, α maps Γ n,1 onto Γ n,1 . Let Γ n,1 = {10 n-1 , 0 n-1 1} and Γ n,1 = Γ n,1 \ Γ n,1
. Since 10 n-1 and 0 n-1 1 are the only vertices of degree n -1, α maps Γ n,1 and Γ n,1 onto Γ n,1 and Γ n,1 , respectively. We distinguish two cases.

Case 1: α(10 n-1 ) = 10 n-1 . Then, because α maps Γ n,1 onto Γ n,1 , we have α(0 n-1 1) = 0 n-1 1. Among the vertices of Γ n,1 , only 010 n-2 has no common up-neighbor with 10 n-1 . Therefore, α(010 n-2 ) = 010 n-2 . In turn, among the remaining vertices of Γ n,1 , only 0010 n-3 has no common up-neighbor with 010 n-2 . Therefore α(0010 n-3 ) = 0010 n-3 . By proceeding with the same argument, α fixes Γ n,1 pointwise and hence fixes Γ n,1 pointwise. Now apply Lemma 2.1 and induction on k to conclude that α fixes Γ n,k pointwise for all k. Therefore α = id in this case.

Case 2: α(10 n-1 ) = 0 n-1 1. Now α(0 n-1 1) = 10 n-1 . Among the vertices of Γ n,1 , only 010 n-2 has no common upneighbor with 10 n-1 . Thus α(010 n-2 ) = 0 n-2 10, which is the only element of Γ n,1 with no common up-neighbor together with α(10 n-1 ) = 0 n-1 1. By proceeding with the same argument, α reverses all the elements of Γ n,1 , that is, α = r on Γ n,1 and consecutively α = r on Γ n,1 . By Lemma 2.1 and induction on k, the same holds for any Γ n,k , k ≥ 2. Therefore α = r in this case.

Let n ≥ 1. An equivalent way to define Λ n is that it is the subgraph of Q n induced on all the binary strings of length n that have no two consecutive 1's in circular manner. This definition is more symmetric than the definition of the Fibonacci strings, so it is reasonable to expect that Aut(Λ n ) is richer than Aut(Γ n ). This is indeed the case. Define

ϕ : Λ n → Λ n by ϕ(b 1 b 2 . . . b n ) = b n b 1 . . . b n-1 . (2)
By the above remark it is clear that ϕ ∈ Aut(Λ n ). Zagaglia Salvi [START_REF] Salvi | The automorphism group of the Lucas semilattice[END_REF] proved that the automorphism groups of the Lucas semilattices are the dihedral groups. The arguments that determine the automorphism group of the Lucas cubes are in a way parallel to the arguments from [START_REF] Salvi | The automorphism group of the Lucas semilattice[END_REF], hence we next give just a sketch of them. Note first that Lemma 2.1 (with the same proof) applies to Lucas cubes as well. Let α ∈ Aut(Λ n ). Suppose that for some a, b ∈ {0, 1, . . . , n -1}, α(10 n-1 ) = 0 a 10 n-a-1 and α(0 n-1 1) = 0 b 10 n-b-1 , where computations are mod n. Then either b = a -1 or b = a + 1 because α(10 n-1 ) and α(0 n-1 1) cannot have a common up-neighbor. When b = a -1 we get α = ϕ a and in the other case α = ϕ a+1 • r. We conclude that Aut(Λ n ) is generated by r and ϕ a for 0 ≤ a ≤ n -1, and hence:

Theorem 2.3 For any n ≥ 3, Aut(Λ n ) D 2n .

The domination number

In this section we consider the domination number of Fibonaci and Lucas cubes. We first interrelate their domination numbers. Then we discuss exact domination numbers for small dimensions. The section is conluded by establishing a general lower bound on the domination number of Lucas cubes.

Proposition 3.1 Let n ≥ 4, then (i) γ(Λ n ) ≤ γ(Γ n-1 ) + γ(Γ n-3 ) , (ii) γ(Λ n ) ≤ γ(Γ n ) ≤ γ(Λ n ) + γ(Γ n-4 ) .
Proof. (i) V (Λ n ) can be partitioned into vertices that start with 0 and vertices that start with 1. The latter vertices are of the form 10 . . . 0 and hence can be dominated by {10b0 | b ∈ U } where U is a minimum dominating set of Γ n-3 with γ(Γ n-3 ) vertices.

While the former vertices can be dominated by γ(Γ n-1 ) vertices. (ii) Let D be a minimum dominating set of Γ n and set

D = {u | u is a Lucas string from D} ∪ {0b 2 . . . b n-1 0 | 1b 2 . . . b n-1 1 ∈ D} .
A vertex 1b 2 . . . b n-1 1 dominates two Lucas vertices, namely 0b 2 . . . b n-1 1 and 1b 2 . . . b n-1 0. Since these two vertices are dominated by 0b 2 . . . b n-1 0, we infer that D is a dominating set of Λ n . It follows that γ(Λ n ) ≤ γ(Γ n ).

A dominating set of Λ n dominates all vertices of Γ n but the vertices of the form 10b 3 . . . b n-2 01. These vertices can be dominated by γ(Γ n-4 ) vertices.

Pike and Zou [START_REF] Pike | The domination number of Fibonacci cubes[END_REF] obtained exact values of γ(Γ n ) for n ≤ 8, see Table 2. By computer search they found 509 minimum dominating sets of Γ 8 . Following their approach we have computed the domination numbers of Λ n , n ≤ 8, see Table 2 again.

Hence the smallest Fibonacci cube and Lucas cube for which the domination numbers are not known are Γ 9 and Λ 9 . Since γ(Γ n ) ≤ γ(Γ n-1 )+γ(Γ n-2 ), it follows that γ(Γ 9 ) ≤ 20, cf. [START_REF] Pike | The domination number of Fibonacci cubes[END_REF]Lemma 3.1]. In order to find a smaller dominating set we have used a local search procedure, that is, to get a new dominating set we have replaced one or more vertices with another vertex. In this way we were able to construct a dominating set of Γ 9 of size 17 given on the left-hand side of Table 1. Similarly we have found a dominating set of Λ 9 of order 16 given on the right-hand side of Table 1. Hence: Proposition 3.2 γ(Γ 9 ) ≤ 17 and γ(Λ 9 ) ≤ 16. 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 To establish the announced lower bound, we will apply the natural concept of overdomination, just as it is done in [START_REF] Pike | The domination number of Fibonacci cubes[END_REF]. It is defined as follows. Let D be a dominating set of a graph G. Then the over-domination of G with respect to D is:

OD G (D) = v∈D (deg G (v) + 1) -|V (G)| . (3)
Note that OD G (D) = 0 if and only if D is a perfect dominating set [START_REF] Livingston | Perfect dominating sets[END_REF][START_REF] Hatami | Perfect dominating sets in the Cartesian products of prime cycles[END_REF], that is, a dominating set such that each vertex is dominated exactly once.

Theorem 3.5 For any n ≥ 7, γ(Λ n ) ≥ L n -2n n -3 . Proof. Let D be a minimum dominating set of Λ n . Set D 1 = D ∩ Λ n,1 and D 2 = D ∩ Λ n,
OD(G) = u∈V (Λn) (|{v ∈ D | d(u, v) ≤ 1}| -1) . (4)
For a vertex

u of Λ n , set t(u) = |{v ∈ D | d(u, v) ≤ 1}| -1.
As D is a dominating set, t(u) ≥ 0 for all u ∈ V (Λ n ). We now distinguish two cases.

Case 1: 0 n ∈ D. Combining Lemma 3.3 with Equation (3) we get

OD(D) ≤ (n + 1) + k(n -1) + l(n -2) + (γ(Λ n ) -k -l -1)(n -3) -L n = γ(Λ n )(n -3) + 2k + l + 4 -L n .
Also as t(u) ≥ 0 for all u ∈ V , Equation (4) implies

OD(D) ≥ t(0 n ) + v∈D 1 t(v) ≥ 2k . Therefore γ(Λ n ) ≥ Ln-l-4 n-3 ≥ Ln-n-4 n-3
.

Case 2: 0 n / ∈ D. Again, combining Lemma 3.3 with Equation (3) we infer

OD(D) ≤ k(n -1) + l(n -2) + (γ(Λ n ) -k -l)(n -3) -L n = γ(Λ n )(n -3) + 2k + l -L n .
Let A be the set of down-neighbors of D 2 . Then for u ∈ D 1 ∩ A, t(u) ≥ 1. By Lemma 3.4, |A| ≥ l and hence |D 1 A| ≥ k + l -n. Therefore by Equation (4),

OD(D) ≥ v∈D 1 A t(v) ≥ k + l -n . Thus γ(Λ n ) ≥ Ln-k-n n-3 ≥ Ln-2n n-3 .
By Case 1 and Case 2, γ(Λ n ) ≥ Ln-2n n-3 .

The 2-packing number

We now turn to the 2-packing number and first prove the following asymptotical lower bound.

Theorem 4.1 For any n ≥ 8, ρ(Γ n ) ≥ ρ(Λ n ) ≥ 2 2 lg n 2 -1 .
Proof. Since for any n ≥ 1, Λ n is an isometric subgraph of Γ n , cf. [START_REF] Klavžar | On median nature and enumerative properties of Fibonacci-like cubes[END_REF], a 2-packing of Λ n is also a 2-packing of Γ n . Therefore ρ(Γ n ) ≥ ρ(Λ n ). Let r, s ≥ 1 and let X and Y be maximum 2-packings of Λ r and Λ s , respectively. Then

{x0y | x ∈ X, y ∈ Y } is a 2-packing of Λ r+s+1 of size ρ(Λ s )ρ(Λ s ). It follows that ρ(Λ r+s+1 ) ≥ ρ(Λ r )ρ(Λ s ) . Set now k = lg n . Then ρ(Λ 2 k ) ≥ ρ(Λ 2 k-1 +1 ) ≥ ρ(Λ 2 k-2 ) 2
. By repeatedly applying this argument we get

ρ(Λ n ) ≥ ρ(Λ 2 k ) ≥ ρ(Λ 2 k-2l ) 2 l . When k is even, take l = k-2 2 to get ρ(Λ n ) ≥ ρ(Λ 4 ) 2 k-2 2 = 2 2 k-2 2 . When k is odd, take l = k-3 2 to get ρ(Λ n ) ≥ ρ(Λ 8 ) 2 k-3 2 ≥ 8 2 k-3 2 = 2 3×2 k-3 2 ≥ 2 2 k-2 2 .
Using computer we obtained the 2-packing numbers of Γ n and Λ n for n ≤ 10 given in Table 2. Table 2: Domination numbers and 2-packing numbers of small cubes Table 2 needs several comments.

• The computer search found exactly ten 2-packings of size 20 in Γ 10 . This already implies that ρ(Γ 10 ) = 20. Indeed, if Γ 10 would contain a 2-packing of size 21, then it would contain at least twenty-one 2-packings of size 20.

• By exhaustive search with computer no 2-packing of size 19 but eighty 2-packing of size 18 in Λ 10 were found, hence ρ(Λ 10 ) = 18.

• There is only one (up to isomorphisms of the graphs considered) maximum 2-packing of Λ 5 , Λ 6 , Λ 7 , Λ 9 , as well as Γ 6 . There are two non-isomorphic 2-packings of maximum cardinality of Γ 9 , which are presented in Table 3.

Since the reverse map given in ( 1) is an automorphism of Fibonacci cubes, the reverse of a 2-packing is also a 2-packing. Interestingly, the maximum 2-packing of Γ 9 shown on the left-hand side of Table 3, denoted X, is also invariant under the reverse map. That is, r(X) = X.

Similarly, the shifts ϕ i , where ϕ is given in (2) and are automorphisms of Lucas cubes, hence they map 2-packings into 2-packings. Now consider the 2-packing of Λ 9 shown in Table 4, denote it Y . Then it can be checked that ϕ 3 (Y ) = Y . As a consequence, ϕ 6 (Y ) = Y . 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 1 Table 3: Maximum 2-packings of Γ 9 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0

Concluding remarks

Based on the date from Table 2 we ask whether some of the followings are true. Problem 5.1 Is it true that (i) γ(Γ n ) -ρ(Γ n ) ≥ γ(Λ n ) -ρ(Λ n ) for n ≥ 1? (ii) γ(Λ n ) ≥ ρ(Γ n ) for n ≥ 4? (iii) γ(Λ n ) ≤ γ(Γ n-1 ) + γ(Γ n-3 ) -1 for n ≥ 6?

Note that the last question, if it has an affirmative answer, reduces the bound of γ(Λ n ) in Proposition 3.1 (i) by 1. Moreover, if (iii) is true, then one can also ask whether γ(Λ n ) ≤ γ(Γ n-1 ) + γ(Γ n-4 ) holds for n ≥ 6.

For a binary string b = b 1 b 2

 2 . . . b n , let b be the binary complement of b and let b R = b n b n-1 . . . b 1 be the reverse of b. For binary strings b and c of equal length, let b + c denote their sum computed bitwise modulo 2. For 1 ≤ i ≤ n, let e i be the binary string of length n with 1 in the i-th position and 0 elsewhere. According to this notation, Γ n,1 = Λ n,1 = {e 1 , e 2 , . . . , e n }.

  2 , and let k = |D 1 | and l = |D 2 |. Then clearly 0 ≤ k, l ≤ n. Note that the over-domination of G with respect to D can be rewritten as

Table 1 :

 1 A dominating set of Γ 9 and a dominating set of Λ 9

Table 4 :

 4 Maximum 2-packing of Λ 9

We conjecture that γ(Γ 9 ) = 17 and γ(Λ 9 ) = 16 hold.

Pike and Zou [START_REF] Pike | The domination number of Fibonacci cubes[END_REF] also proved that for any n ≥ 4,

We next prove a parallel lower bound for the domination number of Lucas cubes. For this sake we first consider degrees of some specific vertices in Lucas cubes.

Let n ≥ 1. Recall that Λ n,1 is the set of all the vertices with exactly one 1. In addition, set

where we again compute by modulo n. Hence Λ n,2 is the subset of Λ n,2 consisting of the Lucas strings containing (in circular manner) 101 as a substring.

Then for the Lucas cube Λ n the followings hold.

(i) The vertex 0 n is the only vertex of the maximum degree n.

(ii) The vertices of Λ n,1 have degree n -2.

(iii) Among the vertices with at least two 1's, only the vertices of Λ n,2 have degree n -3 and all the other vertices have degree at most n -4.

Proof. (i) and (ii) are clear.

(iii) Let u ∈ Λ n,k for some k ≥ 2. Then u has k down-neighbors. The up-neighbors of u are obtained by switching a bit 0 into 1. Let i 1 < i 2 < • • • < i k be the positions in which u contains 1. Throughout the proof, the indices of i's will be considered by modulo k and i j by modulo n. As no consecutive bits of 1's are allowed, i j+1 -i j ≥ 2 for all 1 ≤ j ≤ k. Let I j = {i j -1, i j + 1} be the set of the positions which are adjacent to i j for each 1 ≤ j ≤ k and let I = 1≤j≤k I j . Then any bit which is not in I can be switched to 1 and hence the number of up-neighbors of u is n -k -|I|. Therefore, deg(u) = n -|I|. Note that I j ∩ I j = ∅ if |j -j | ≥ 2, therefore by pigeon-hole principle, |I| ≥ k. The equality holds if and only if I j I j+1 = ∅ for all 1 ≤ j ≤ k, which occurs if and only if i j+1 = i j + 2 for all 1 ≤ j ≤ k, which is in turn if and only if n is even and k = n Proof. For 1 ≤ i ≤ , let A i be the set of down-neighbors of some v i ∈ Λ n,2 . Then |A i | = 2 for each i. Considering bits by modulo n, each vertex 0 a 10 n-a-1 in Λ n,1 can be a down-neighbor of at most two vertices 0 a 1010 n-a-3 and 0 a-2 1010 n-a-1 , and hence at most two of v 1 , . . . , v l . By pigeon-hole principle, the assertion is true.