SPANNING SIMPLICIAL COMPLEXES OF n-cyclic GRAPHS WITH A COMMON EDGE. Guangjun Zhu, Feng Shi and Yuxian Geng

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1 Iteratioal Electroic Joural of Algera Volume SPANNING SIMPLICIAL COMPLEXES OF -CYCLIC GRAPHS WITH A COMMON EDGE Guagju Zhu, Feg Shi ad Yuxia Geg Received: 1 August 2013; Revised: 5 Novemer 2013 Commuicated y Sait Halıcıoğlu Dedicated to the memory of Professor Efraim P Armedariz Astract I this paper, we characterize some algeraic ad comiatorial properties of spaig simplicial complex sg t1, t 2,, t of the class of the -cyclic graphs G t1, t 2,, t with a commo edge We show that sg t1, t 2,, t is pure simplicial complex of dimesio t i 2, ad we also determie the Staley-Reiser ideal I sgt1, t 2,, t of sg t1, t 2,, t ad its primary decompositio Uder the coditio that the legth of every cyclic graph G ti is t for 1 i, we give a formula for f-vector of sg t1, t 2,, t ad cosequetly a formula for Hilert series of the Staley- Reiser rig k[ sg t1, t 2,, t, where k is a field Mathematics Suject Classificatio 2010: 13P10, 13H10, 13F20, 13C14 Keywords: Spaig tree, simplicial complexes, f-vector, h-vector, Hilert series 1 Itroductio The ote of spaig simplicial complex s G o edge set E of a graph G = GV, E was itroduced i [1, the set of its facets is exactly edge set sg of all possile spaig trees of G, ie s G = F i F i sg Note that for a graph G, the prolem of fidig sg is ot always easy to hadle Awar, Raza ad Kashif [1 proved some algeraic ad comiatorial properties of spaig simplicial complexes of the ui-cyclic graph U, where U is a coected graph o vertices, which cotais exactly oe cycle of legth I this paper, our goal is to characterize some algeraic ad comiatorial properties of spaig simplicial complexes of a class of -cyclic graphs G t1, t 2,, t with a commo edge, which is otaied y joiig cyclic graphs G t1, G t2,, G t of legth t 1, t 2,, t with a commo edge For = 2 ad t 1 = t 2 = 4, the graph of G 4, 4 is show i Figure 1 of Sectio 2 This research is partially supported y the Natioal Natural Sciece Foudatio of Chia , the Natural Sciece Foudatio of Jiagsu provicebk , ad the Natural Sciece Foudatio for Colleges ad Uiversities i Jiagsu Provice 10KJB110007; 11KJB110011

2 SPANNING SIMPLICIAL COMPLEXES OF -CYCLIC GRAPHS 133 We give a rief overview of this paper I Sectio 2, we recall some defiitios ad results from commutative algera ad algeraic comiatorics I Sectio 3, we determie the Staley-Reiser ideal I sg t1, t 2,, t of s G t1, t 2,, t ad its primary decompositio i Theorem 32 I Sectio 4, uder the assumptio that the legth of cyclic graph G ti is t for every 1 i, we give a formula for f-vector of s G t1, t 2,, t ad cosequetly a formula for Hilert series of the Staley-Reiser rig k[ s G t1, t 2,, t, where k is a field 2 Prelimiaries We firstly recall some defiitios ad asic facts aout graph ad simplicial complex to make this paper self-cotaied Defiitio 21 A spaig tree of a simple coected fiite graph G = GV, E is a sugraph of G, which is a tree ad cotais all vertices of G We deote the collectio of all edge sets of the spaig trees of G y sg, ie See [3 for more details sg = {ET i E T i is a spaig tree of G} It is well kow that for ay simple coected fiite graph, spaig trees always exist Oe ca fid a spaig tree systematically y the cuttig-dow method, which says that a spaig tree is otaied y removig oe edge from each cycle appearig i the graph For example, for the followig graph G, we otai that sg = {{e 2, e 3, e 5, e 6, e 7 }, {e 1, e 3, e 5, e 6, e 7 }, {e 1, e 2, e 5, e 6, e 7 }, {e 1, e 2, e 3, e 5, e 6 }, {e 2, e 3, e 4, e 6, e 7 }, {e 1, e 3, e 4, e 6, e 7 }, {e 1, e 2, e 4, e 6, e 7 }, {e 2, e 3, e 4, e 5, e 7 }, {e 1, e 3, e 4, e 5, e 7 }, {e 1, e 2, e 4, e 5, e 7 }, {e 1, e 2, e 4, e 5, e 6 }, {e 1, e 3, e 4, e 5, e 6 }, {e 2, e 3, e 4, e 5, e 6 }, {e 1, e 2, e 3, e 4, e 6 }, {e 1, e 2, e 3, e 4, e 5 }} e 1 e 6 e 2 e 7 e 5 e 3 e 4 Figure 1 2-cyclic graph with a commo edge

3 134 GUANGJUN ZHU, FENG SHI AND YUXIAN GENG Defiitio 22 A simplicial complex o a set of vertices [ = {1, 2,, } is a collectio of susets of [ such that 1 {i} for each {i} [; 2 if F ad G F, the G A elemet of is called a face of, ad the dimesio of a face F of is defied as F 1, where F is the umer of vertices of F ad deoted y dim F The faces of dimesio 0 ad 1 are called vertices ad edges, respectively, ad dim = 1 The maximal faces of uder iclusio are called facets of The dimesio of the simplicial complex, which is deoted y dim, is the maximal dimesio of its facets, ie dim = max {dim F F is a facet of } We deote the simplicial complex with facets {F 1,, F q } y = F 1,, F q Defiitio 23 A simplicial complex is pure if all of its facets have the same dimesio Defiitio 24 Give a simplicial complex of dimesio d, we defie its f-vector to e the d 1-tuple f = f 0, f 1,, f d, where f i is the umer of i-dimesioal faces of Defiitio 25 For a simple coected fiite graph G = GV, E with sg = {E 1,, E s }, we defie a simplicial complex s G o E such that facets of s G are precisely the elemets of sg, called the spaig simplicial complex of GV, E I other words, s G = E 1,, E s For example, the spaig simplicial complex of the graph G with edge set E = {e 1, e 2, e 3, e 4, e 5, e 6, e 7 } i Figure 1 is give y s G = {e 2, e 3, e 5, e 6, e 7 }, {e 1, e 3, e 5, e 6, e 7 }, {e 1, e 2, e 5, e 6, e 7 }, {e 1, e 2, e 3, e 5, e 6 }, {e 2, e 3, e 4, e 6, e 7 }, {e 1, e 3, e 4, e 6, e 7 }, {e 1, e 2, e 4, e 6, e 7 }, {e 2, e 3, e 4, e 5, e 7 }, {e 1, e 3, e 4, e 5, e 7 }, {e 1, e 2, e 4, e 5, e 7 }, {e 1, e 2, e 4, e 5, e 6 }, {e 1, e 3, e 4, e 5, e 6 }, {e 2, e 3, e 4, e 5, e 6 }, {e 1, e 2, e 3, e 4, e 6 }, {e 1, e 2, e 3, e 4, e 5 } Defiitio 26 A -cyclic graph G t1, t 2,, t with a commo edge is a coected graph havig t i 2 1 vertices ad t i 1 edges, otaied y joiig cyclic graphs G t1, G t2,, G t with a commo edge, where G ti deotes the cyclic

4 SPANNING SIMPLICIAL COMPLEXES OF -CYCLIC GRAPHS 135 graph of legth t i We ca assume that t 1 t 2 t ad t i 3 for each i {1, 2,, } 3 Primary decompositio of the Staley-Reiser ideal I this sectio, we will determie the Staley-Reiser ideal I sg t1, t 2,, t of s G t1, t 2,, t ad its primary decompositio We lael the edge set of G t1, t 2,, t such that {e i1, e i2,, e i ti } is the edge set of cyclic graph G ti for every 1 i By covetio, e 1 t1 = e 2 t2 = = e t = e is the commo edge First, we have the followig propositio Propositio 31 s G t1, t 2,, t is a pure simplicial complex of dimesio t i 2 Proof Let E = {e 11, e 12,, e 1, t11, e 21,, e 2, t21,, e 1,, e, t1, e} e the edge set of -cyclic graph G t1, t 2,, t, where e is the commo edge As G t1, t 2,, t cotais exactly cycles of legth t 1, t 2,, t, which has a commo edge e, y the cuttig-dow method, its spaig trees are otaied y removig oe edge from each cycle G ti, 1 i Hece, the suset ET i E is i sg t1, t 2,, t if ad oly if ET i = E\{e 1 i1, e 2 i2,, e i } for some i j {1, 2,, t j }, where these e j, ij s are distict ad j rus from 1 to, with covetio e 1 t1 = e 2 t2 = = e t = e, ie sg t1, t 2,, t ={E\{e 1 i1,, e i } 1 i j t j, 1 j, where e j, ij s are distict} It is easily see that each spaig tree of s G t1, t 2,, t has t i 1 = t i 2 1 edges Thus the result follows Let E = {e 11, e 12,, e 1, t11, e 21,, e 2, t21,, e 1,, e, t1, e} e the edge set of -cyclic graph G t1, t 2,, t, ad let s G t1, t 2,, t e the spaig simplicial complex of G t1, t 2,, t We ca assume that S = k[x 11,, x 1, t11, x 21,, x 2, t21,, x 1,, x, t1, y is a polyomial rig i t i 1 variales over a field k, I sg t1, t 2,, t is the Staley-Reiser ideal of s G t1, t 2,, t, which is a square-free moomial ideal The stadard graded algera k[ s G t1, t 2,, t = S/I sg t1, t 2,, t is called the Staley-Reiser rig of s G t1, t 2,, t We ca give a primary decompositio of ideal I sg t1, t 2,, t, Hilert series ad h-vector of k[ s G t1, t 2,, t We refer readers to [2 ad [5 for detailed iformatio aout the Staley-Reiser ideal, primary decompositio, Hilert series ad h-vector Now, we give a primary decompositio of the Staley-Reiser ideal I sg t1, t 2,, t of s G t1, t 2,, t

5 136 GUANGJUN ZHU, FENG SHI AND YUXIAN GENG Theorem 32 Let s G t1, t 2,, t e the spaig simplicial complex of -cyclic graph G t1, t 2,, t The the Staley-Reiser ideal I sg t1, t 2,, t of s G t1, t 2,, t is give y I sg t1, t 2,, t = i j {1,, t j 1} j {1,, } 11 = y x 1i1, x 2i2,, x i 21 x 1j, y i j {1,, t j 1} j {1,, ˆk,, } k {1,, } 1 x 2j,, y x j, x 1i1,, ˆx kik,, x i, y 1 i < j 1 s t i 1 1 l t j 1 x is x j l Proof As each of facets of s G t1, t 2,, t is otaied y removig exactly oe edge from each cycle G ti, 1 i, from [5, Propositio 5310, we get that I sg t1, t 2,, t = i j {1,, t j 1} j {1,, } x 1i1,, x i i j {1,, t j 1} j {1,, ˆk,, } k {1,, } x 1i1,, ˆx kik,, x i, y From [4, Propositio 121, we have that i j {1,, t j 1} j {1,, } x 1i1, x 2i2,, x i = x 11, x 21,, x 1,1, x 1 x 11,, x 1,1, x 2 x 11, x 21,, x 1,1, x, t1 x 11, x 21,, x 1, 2, x 1 x 11,, x 1, 2, x, t1 x 1, t11, x 2, t21,, x 1, t1 1, x, t1 1 1 = x 11, x 21,, x 1,1, x j x 11, x 21,, x 1, 2, x j 1 x 1, t11, x 2, t21,, x 1, t1 1, x j = x 11, x 21,, = x 1j, t 1 1 x 2j,, 1 x 1, j, t 1 1 x j x 1, t11, x 2, t21,, 1 x 1, j, x j, t x 1, j, x j

6 SPANNING SIMPLICIAL COMPLEXES OF -CYCLIC GRAPHS 137 ad i j {1,, t j 1} j {1,, ˆk,, } k {1,, } x 1i1,, ˆx kik,, x i, y= x 11, x 21,, x 1,1, y x 11, x 21,, x 1, 2, y x 11, x 21,, x 1, t1 1, y x 11, x 21,, y, x 1 x 11, x 21,, y, x 2 x 11, x 21,, y, x, t1 y, x 2, t21,, x, t1 = x 11, x 21,, = x 1j, Therefore, t 1 1 x 2j,, 1 x 1, j, y x 11, x 21,, y, t y, x 2j,, x j I sg t1, t 2,, t = = x 1j, = y 21 x 1j, i j {1,, t j 1} j {1,, } x 2j,, 21 x 1j, y t x 2j,, y, 11 x 1, j, y x 1i1, x 2i2,, x i 1 x 1, j, 1 x 2j,, y 21 x 1j, 21 x j y, 1 x 2j,, y, i j {1,, t j 1} j {1,, ˆk,, } k {1,, } 11 x j 21 x j y, x j, 1 i < j 1 s t i 1 1 l t j 1 x is x j l 21 x 1j, 1 x 2j,, x 2j,, x, t1 x j x 1i1,, ˆx kik,, x i, y x 2j,, x j t 1 1 x 1, j, y 4 The computatio of f-vector of s G t1, t 2,, t I this sectio, we will give a formula for f-vector of s G t1, t 2,, t ad cosequetly a formula for Hilert series of the Staley-Reiser rig k[ s G t1, t 2,, t uder the assumptio that the legth of every cyclic graph G ti is t for 1 i But efore this we eed the followig propositio, its proof ca e see i Propositio 22 of [1 Propositio 41 For a simplicial complex o [ of dimesio d, if f t = t1 for some t d, the f i = i1 for all 0 i < t

7 138 GUANGJUN ZHU, FENG SHI AND YUXIAN GENG Now, uder the assumptio that the legth of the cyclic graph G ti is t for every 1 i, we give the formula to compute the f-vector of s G t1, t 2,, t Theorem 42 Let t i = t for every 1 i, ad = t 1 1 The the f-vector of s G t1, t 2,, t is give y f = f 0, f 1,, f d, where d = t 2 ad 4 1 f j = i i1 i j1 j1 t 1 jt1 j1 t 1 jt1 2t2 2 j2t21 2 j1 1 i [iti1 i j[iti11 2t1 2 j2t21 2 j1 1 i [iti1 i j[iti i i1 [iti1 i jiti1 3 j1 1 i [iti1 i j[iti i i1 [iti1 i jiti1 3 j1 1 i [iti1 i j[iti11 [iti1 jiti1 j1 m 1 i m 1 i i1 j1 m1 m 1 i 1 i i1 j1 2 1 i 1 i i1 1 j1 1 1 i 1 i i1 1 if 0 j t 2, if t 1 j mi{2t3, d1}1, if j = mi{2t2, d1}1, [iti1 i j[iti11 [iti1 i jiti1 [iti1 i j[iti11 [iti1 i jiti1 [iti1 i j[iti11 [iti1 i jiti1 [iti1 i j[iti11 [iti1 i jiti1 if mi{2t2, d1} j mi{3t4, d1}1, if j = mi{3t3, d1}1, if mi{3t3, d1} j mi{4t5, d1}1, if j = mi{4t4, d1}1, if mi{mt1, d1} j mi{m1t11, d1}1, if j = mi{m1t1, d1}1, if j =mi{1t1, d1}1, if mi{1t1, d1} j d

8 SPANNING SIMPLICIAL COMPLEXES OF -CYCLIC GRAPHS 139 Proof Let E = {e 11, e 12,, e 1, t1, e 21,, e 2, t1,, e 1,, e, t1, e} e the edge set of -cyclic graph G t1, t 2,, t, where e is the commo edge By the defiitio of f-vector of s G t1, t 2,, t, f j is the umer of all those susets of the edge set E of graph G t1, t 2,, t, with j 1 elemets, that cotai either {e i1, e i2,, e i, t1, e} for all 1 i or {e k1, e k2,, e k, t1, e l1, e l 2,, e l, t1 } for 1 k < l Take ay suset F E cosistig of t 1 elemets For every 1 j the edge set {e j1,, e j, t1, e} of the cyclic graph G tj has t elemets, it is clear that {e j1, e j 2,, e j, t1, e} ca ot appear i F, so F s G t1, t 2,, t It follows that s G t1, t 2,, t cotais all possile susets of E with cardiality t 1, therefore f t2 = t1 t1 = t1 Thus, y Propositio 41, we have fj = j1 for all 0 j t 2 For t 1 j mi{2t 3, d 1} 1, we eed to cout all the susets E with cardiality j 1 cotaiig the edge set {e j1, e j 2,, e j, t1, e} of some G tj of the -cyclic graph G t1, t 2,, t The edge set E of the graph G t1, t 2,, t has = t 1 1 elemets, ad there are t 1 jt1 susets of E with cardiality j 1 such that {e j1, e j 2,, e j, t1, e} is a part of it I total, there are j1 susets of E with cardiality j 1, hece f j = j1 t 1 jt1 Whe j = mi{2t 2, d 1} 1, we eed to compute all the susets of E havig the cardiality j 1 cotaiig such edge sets {e k1,, e k, t1, e l1,, e l, t1 } for 1 k < l of some two cyclic graphs G tk ad G tl of -cyclic graph G t1, t 2,, t, we get that there are 2t2 2 j2t21 such susets It is clear that we also eed to compute these susets of E havig the cardiality j 1 cotaiig such edge set {e i1, e i 2,, e i, t1, e} of some cyclic graph G ti of -cyclic graph G t1, t 2,, t we get that there are t 1 jt1 such susets I total, we have j1 susets of E with cardiality j 1 Therefore, f j = j1 t 1 jt1 2t2 2 j2t21 For mi{2t 2, d 1} j mi{3t 4, d 1} 1, o the oe had, we eed to cout all the susets of E havig the cardiality j 1 cotaiig such edge set {e i1, e i 2,, e i, t1, e j1, e j 2,, e j, t1, e} of some two cyclic graphs G ti ad G tj of -cyclic graph G t1, t 2,, t, y the iclusio exclusio priciple, we get there are 2t1 2 j2t11 such susets O the other had, we eed to compute all the susets of E havig the cardiality j 1 cotaiig such edge set {e i1, e i 2,, e i, t1, e} of some cyclic graph G ti of -cyclic graph G t1, t 2,, t, we get that there are 1 [ t jt1 1 2t1 1 j2t11 such susets It is ovious that we also eed to compute all the susets of E havig the cardiality j 1 cotaiig {e k1, e k2,, e k, t1, e l1, e l 2,, e l, t1 } for 1 k < l, there are 2 [ 2t2 j2t21 2t1 j2t11 = 2t1 2 j2t21 such susets Therefore f j = j1 2t1 2 j2t11 1 [ t jt1 1 2t1 1 j2t11 2t1 2 j2t21 = 2 j1 1 i [iti1 i j[iti11 2t1 2 j2t21 Whe j = mi{3t 3, d 1} 1, o the oe had, we ot oly eed to compute all the susets of E havig the cardiality j 1 cotaiig such edge set

9 140 GUANGJUN ZHU, FENG SHI AND YUXIAN GENG {e k1, e k2,, e k, t1, e l1, e l 2,, e l, t1, e s1, e s 2,, e s, t1 } for 1 k < l < s of some three cyclic graphs G tk, G tl ad G ts of -cyclic graph G t1, t 2,, t, we get that there are 3 [ 3t3 j3t31 3t2 j3t21 = 3t2 3 j3t31 such susets we also eed to compute all the susets of E havig the cardiality j 1 cotaiig {e k1, e k2,, e k, t1, e l1, e l 2,, e l, t1 } for 1 k < l, y the iclusio exclusio priciple, we otai that there are 2 [ 2t1 j2t21 2 3t2 1 j3t31 such susets O the other had, we ot oly eed to cout all the susets of E havig the cardiality j 1 cotaiig {e k1, e k2,, e k, t1, e l1, e l 2,, e l, t1 } for 1 k < l, ut also eed to compute all the susets of E havig the cardiality j 1 cotaiig {e k1, e k2,, e k, t1, e} for 1 k By the iclusio exclusio priciple, there are 2t1 2 j2t11 ad 1 [ t jt1 1 2t1 1 j2t11 such susets respectively There are j1 susets of E with cardiality j1 i total Hece, y the use of repetitio of comiatorial formula k 1 j m = 1 k m1, we have that 2t 1 2 3t 2 f j = [ j 1 2 j 2t j 3t 3 1 3t 2 t 1 2t 1 [ 3 j 3t j t 1 1 j 2t 1 1 2t 1 2 j 2t [it i 1 2t 1 = 1 i j 1 i j [it i j 2t t 2 [ j 3t [it i 1 2t 1 = 1 i j 1 i j [it i j 2t t 2 [ j 3t [it i 1 2t 1 = 1 i j 1 i j [it i j 2t t j 3t [it i 1 = 1 i j 1 i j [it i 1 1 j=0 j k 3 1 i i 1 i [it i 1 j it i 1

10 SPANNING SIMPLICIAL COMPLEXES OF -CYCLIC GRAPHS 141 For mi{3t 3, d 1} j mi{4t 5, d 1} 1, o the oe had, we ot oly eed to cout all susets of E havig the cardiality j 1 cotaiig {e k1, e k2,, e k, t1, e l1, e l 2,, e l, t1, e s1, e s 2,, e s, t1, e} for 1 k < l < s of some three cyclic graphs G tk, G tl ad G ts of -cyclic graph G t1, t 2,, t, ut also eed to cout all susets of E havig the cardiality j 1 cotaiig {e i1, e i 2,, e i, t1, e j1, e j 2,, e j, t1, e} of some two cyclic graphs G ti ad G tj By the iclusio exclusio priciple, there are 3t2 3 j3t21 ad 2 [ 2t1 j2t11 2 3t2 1 j3t21 such susets respectively O the other had, we eed to compute all the susets of E havig the cardiality j1 cotaiig {e i1, e i 2,, e i, t1, e} of some cyclic graph G ti, we get that there are 1 { t jt1 1 1 [ 2t1 j2t11 2 3t2 1 j3t21 1 3t2 2 j3t21 } such susets Of course, we have to cout all the susets E with cardiality j 1 cotaiig {e k1, e k2,, e k, t1, e l1, e l 2,, e l, t1, e s1, e s 2,, e s, t1 } for 1 k < l < s, ad all susets E with cardiality j 1 cotaiig such edge set {e k1, e k2,, e k, t1, e l1, e l 2,, e l, t1 } for 1 k < l By the iclusio exclusio priciple, there are 3 [ 3t3 j3t31 3t2 j3t21 = 3t2 3 j3t31 ad 2 [ 2t1 j2t21 2 3t2 1 j3t31 I total, there are j1 susets of E with cardiality j 1 By the use of repetitio of the comiatorial formula m1 1 j m j = 1 m1, we have that j=0 3t 2 f j = j 1 3 j 3t 2 1 2t 1 2 3t 2 [ 2 j 2t j 3t 2 1 t 1 2t 1 2 3t 2 { [ 1 j t 1 1 j 2t j 3t t 2 3t 2 } 2 j 3t j 3t 3 1 2t 1 2 3t 2 [ 2 j 2t j 3t 3 1 t 1 2t 1 = [ j 1 1 j t j 2t t 2 [ j 3t 2 1 2t 1 2 3t 2 [ 2 j 2t j 3t 3 1

11 142 GUANGJUN ZHU, FENG SHI AND YUXIAN GENG t 2 2 2t 1 = [ j 1 1 j t j 2t t 2 { [ } j 3t [it i 1 1 i i 1 i j it i 1 t 2t 1 = j 1 1 j t 1 2 j 2t t 2 [ j 3t [it i 1 1 i i 1 i j it i 1 t 2t 1 3t 2 = j 1 1 j t 1 2 j 2t j 3t [it i 1 1 i i 1 i j it i 1 3 [it i 1 = 1 i j 1 i j [it i [it i 1 1 i i 1 i j it i 1 Other cases ca e show i a similar way as the aove We ca ow give a formula for Hilert series of k[ s G t1, t 2,, t uder the coditio that the legth of every cyclic graph G ti is t for 1 i Theorem 43 Let s G t1, t 2,, t e the spaig simplicial complex of -cyclic graph G t1, t 2,, t, where t i = t for each 1 i The Hilert series of the Staley-Reiser rig k[ s G t1, t 2,, t is give y t2 j1 z i1 2t4 Hk[ s G t1, t 2,, t, z=1 1 z i1 [ j1 t 1 jt1 z i1 1 z i1 { 2 j1 1 i i 1 z 2t2 i=0 [iti1 j[iti11 }z 2t2 i=t1

12 SPANNING SIMPLICIAL COMPLEXES OF -CYCLIC GRAPHS 143 3t5 t2 [ 2 j1 1 i { 2 j1 1 i i { 2 j1 1 i i t12 i=1t1 i [iti1 j[iti z i1 [iti1 j[iti11 [iti1 j[iti11 2t1 j2t i i1 1 z 3t3 1 1 i i1 z i1 [iti1 i jiti1 [iti1 i jiti1 }z 3t3 }z 1t1 1 z 1t1 [ 1 j1 1 i [iti1 1 i j[iti11 1 i i1 [iti1 i jiti1 z i1 1 z i1 Proof From [5, Corollary 545, we have that if is a simplicial complex ad f = f 0,, f d is its f-vector, the the Hilert series of Staley-Reiser rig k[ is give y Hk[, z = d i=1 f i z i1, d = dim 1 z i1 The desired formula follows from the aove theorem at oce Ackowledgmet The authors are grateful to Professor Zhogmig Tag for useful discussios They would like to express their sicere thaks to the editor for help ad ecouragemet Special thaks are due to the referee for a careful readig ad pertiet commets Refereces [1 I Awar, Z Raza ad A Kashif, Spaig simplicial complexes of ui-cyclic graphs, to appear i Algera Colloq 2014 [2 W Brus ad J Herzog, Cohe-Macaulay Rigs, Revised Editio, Camridge Studies i Advaced Mathematics, 39, Camridge Uiversity Press, Camridge, 1998 [3 F Harary, Graph Theory Readig, MA: Addiso-Wesley, 1994 [4 J Herzog ad T Hii, Moomial Ideals, Spriger-Verlag New York Ic, 2011 [5 R H Villarreal, Moomial Algeras, Dekker, New York, 2001

13 144 GUANGJUN ZHU, FENG SHI AND YUXIAN GENG Guagju Zhu ad Feg Shi School of Mathematics Sciece Soochow Uiversity Suzhou , Chia s: G Zhu F Shi Yuxia Geg School of Mathematics ad Physics Jiagsu Teachers Uiversity of Techology Chagzhou , Chia yxgeg@jstueduc