Discrete Mathematics. Laplacian spectral characterization of some graphs obtained by product operation

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1 Dscrete Mathematcs 31 (01) Contents lsts avalable at ScVerse ScenceDrect Dscrete Mathematcs journal homepage: Laplacan spectral characterzaton of some graphs obtaned by product operaton Jang Zhou, Changjang Bu Department of Appled Mathematcs, College of Scence, Harbn Engneerng Unversty, Harbn , PR Chna a r t c l e n f o a b s t r a c t Artcle hstory: Receved 5 August 011 Receved n revsed form 3 November 011 Accepted February 01 Avalable onlne 5 February 01 Keywords: Laplacan spectrum Cospectral graphs Cospectral mate Spectral characterzaton A graph s sad to be DLS, f there s no other non-somorphc graph wth the same Laplacan spectrum. Let G be a DLS graph. We show that G K r s DLS f G s dsconnected. If G s connected, t s proved that G K r s DLS under certan condtons. Applyng ths result, we prove that G K r s DLS f G s a tree on n (n 5) vertces or a uncyclc graph on n (n > 6) vertces. 01 Elsever B.V. All rghts reserved. 1. Introducton All graphs consdered here are smple and undrected. For a graph G, let A(G) be the adjacency matrx of G, let D(G) be the dagonal matrx of vertex degrees of G. The matrx L(G) = D(G) A(G) s called the Laplacan matrx of G. The egenvalues of L(G) are called the Laplacan egenvalues of G. Snce L(G) s real, symmetrc and postve semdefnte, the Laplacan egenvalues of G are all nonnegatve real numbers. The largest egenvalue of L(G) s called the L-ndex of G. It s well-known that the smallest Laplacan egenvalue of G s always 0. The multset of the egenvalues of L(G) s called the Laplacan spectrum of G. Two graphs are sad to be L-cospectral, f they have the same Laplacan spectrum. A graph s sad to be determned by the Laplacan spectrum, f there s no other non-somorphc graph wth the same Laplacan spectrum. We shall use DLS as an abbrevaton for determned by the Laplacan spectrum n ths paper. For two dsjont graphs G and H, let G H denote the dsjont unon of G and H, and mg denote the dsjont unon of m copes of G. Let G denote the complement of G. The product of G and H, denoted by G H, s the graph obtaned from G H by jonng each vertex of G to each vertex of H. Clearly, G H = G H. As usual, P n, C n and K n stand for the path, the cycle and the complete graph on n vertces, respectvely. In partcular, K 1 stands for an solated vertex. Let K 1,n 1 denote the star on n vertces. Whch graphs are determned by ther spectra s a dffcult problem n the theory of graph spectra. Only some graphs wth specal structures have been proved to be determned by ther spectra [4,5,1,14,5]. Some DLS graphs can be obtaned from the product of a DLS graph and an solated vertex or a complete graph. Here we ntroduce some relevant results. (a) Paths and cycles are DLS. The dsjont unon of paths s DLS, and the dsjont unon of cycles s also DLS (see [19]). (b) The mult-fan graph (P n1 P n P ns ) K 1 s DLS (see [1]). (c) The wheel graph C n K 1 s DLS when n 6 (see [4]). (d) The graph C n K m s DLS when n 6, and the graph (P n1 P n P ns ) K m s DLS (see [10]). Correspondng author. E-mal addresses: zhoujang @163.com (J. Zhou), buchangjang@hrbeu.edu.cn (C. Bu) X/$ see front matter 01 Elsever B.V. All rghts reserved. do: /j.dsc

2 159 J. Zhou, C. Bu / Dscrete Mathematcs 31 (01) Let G be a DLS graph. We show that G K m s DLS when G s dsconnected. If G s connected, t s proved that G K n s DLS under certan condtons. Applyng ths result, we prove that G K n s DLS f G s a tree on n (n 5) vertces or a uncyclc graph on n (n > 6) vertces.. Prelmnares In order to get our man results, some helpful lemmas are gven n ths secton. Lemma.1 ([1]). Let µ 1 µ µ n = 0 and µ 1 µ µ n = 0 be the Laplacan spectra of G and G, respectvely. Then µ + µ n = n for any {1,,..., n 1}. Lemma. ([6]). Let G be a connected graph on n vertces, the L-ndex of G s µ(g). Then µ(g) n, wth equalty f and only f G s dsconnected. It s not dffcult to obtan the followng lemma from Lemma.1. Lemma.3 ([4]). Let G 1 and G be graphs wth n 1 and n vertces, respectvely. Let µ 1 µ µ n1 = 0 and η 1 η η n = 0 be the Laplacan spectra of G 1 and G, respectvely. Then n 1 + n, µ 1 + n, µ + n,..., µ n1 1 + n, η 1 + n 1, η + n 1,..., η n 1 + n 1, 0 are the Laplacan egenvalues of graph G 1 G. Lemma.4 ([0]). A graph G s DLS f and only f ts complement G s DLS. Lemma.5 ([0]). Let G be a graph on n vertces wth L-ndex n. If G s DLS, then G mk 1 s DLS for any postve nteger m. The second smallest Laplacan egenvalue of graph G s called the algebrac connectvty of G, denoted by a(g). It s wellknown that G s connected f and only f a(g) > 0. Let κ(g) denote the vertex connectvty of G. Lemma.6 ([9]). Let G be a non-complete, connected graph on n vertces. Then a(g) = κ(g) = 1 f and only f G = H K 1, where H s a dsconnected graph on n 1 vertces. Lemma.7 ([3]). Let G be a connected graph wth vertex set V(G), let µ(g) be the L-ndex of G. Then µ(g) max{d(v) + d(v)m(v) v V(G)}, where d(v) s the degree of vertex v, m(v) s the average degree of all neghbours of vertex v. Lemma.8 ([10]). Let G be a graph. For the Laplacan matrx, the followng nvarants of G can be obtaned from the spectrum: (1) the number of vertces; () the number of edges; (3) the number of components. Lemma.9. Let f be a postve nteger such that f max{f 0, f 1,..., f r } f r, f r 5. Let Γ = r (f r )(f r 3) Γ + 3, = f 0 + f f r, where r, f 0, f 1,..., f r are postve ntegers, and, then f (f 1) wth equalty f and only f {f 0, f 1,..., f r } = {f r, 3, 1,..., 1}. Proof. It s easy to see that f (f 1) Γ = = 1 f + 1 f. Obvously Γ s maxmal r f and only f f s maxmal. Wthout loss of generalty, let f 0 = max{f 0, f 1,..., f r }. Frst we r wll show that f s not maxmal when f 0 < f r. If f 0 < f r, by f 0 + f f r = f, t s easy to see that there exsts a postve nteger j (1 j r) such that f j > 1. If we replace f 0 and f j wth f and f j 1 respectvely, then the sum of the squares ncreases. So r f s not maxmal when f 0 < f r. If f 0 = f r, then f 1 +f + +f r = r r +. It s not dffcult to see that f s maxmal f and only f {f 1, f,..., f r } = {3, 1, 1,..., 1}. Hence we have Γ = r f (f 1) (f r )(f r 3) + 3, wth equalty f and only f {f 0, f 1,..., f r } = {f r, 3, 1,..., 1}. Lemma.10. Let G be a star. Then G K r s DLS for any postve nteger r.

3 J. Zhou, C. Bu / Dscrete Mathematcs 31 (01) Proof. Suppose that G = K 1,n 1. By Lemma.4 we know that K 1,n 1 K r s DLS f and only f K n 1 (r + 1)K 1 s DLS. Snce K n 1 s a DLS graph wth L-ndex n 1, by Lemma.5, K n 1 (r + 1)K 1 s DLS. Hence K 1,n 1 K r s DLS. Let K n e denote the graph obtaned from complete graph K n by deletng one edge. Let G be a graph L-cospectral wth K n e. Lemma.8 mples that G s a connected graph on n vertces and n(n 1) 1 edges. Hence G = K n e,.e., K n e s DLS. Lemma.11. Let G = (K (n 3)K 1 ) K 1 (n 6). Then G K r s DLS for any postve nteger r. Proof. By Lemma.4 we know that G K r s DLS f and only f (K n 1 e) (r + 1)K 1 s DLS. The L-ndex of K n 1 e s n 1. Snce K n 1 e s DLS, by Lemma.5, (K n 1 e) (r + 1)K 1 s DLS,.e., G K r s DLS. Lemma.1 ([10]). The graph C n K m s DLS when n 6. Lemma.13 ([]). The graph K a K b (b > 1) wth a > 5 s DLS. b 3 Lemma.14 ([6]). Let G be a graph on n vertces, and the Laplacan egenvalues of G are µ 1 µ µ n = 0. Then the number of spannng trees of G s 1 n 1 µ n =1. 3. Man results For a dsconnected DLS graph G, t s known that the product G K 1 s DLS (cf. [0, Proposton 4]). Ths property also holds f K 1 s replaced by a complete graph. Theorem 3.1. Let G be a dsconnected DLS graph on n vertces, then G K m s DLS for any postve nteger m. Proof. By Lemma.4 we know that G K m s DLS f and only f G mk 1 s DLS. Snce G s dsconnected, ts complement G s connected. By Lemma., the L-ndex of G s n. Snce G s DLS, by Lemma.4 we know that G s DLS. Lemma.5 mples that G mk 1 s DLS. Hence G K m s DLS. Remark 3.1. Snce the dsjont unon of paths s DLS (see [19]), by Theorem 3.1, we know that graph (P n1 P n P ns ) K m s DLS. In [10], ths result s obtaned by nducton on m. Some dsconnected DLS graphs can be found n [,16,17,7,]. For a connected graph G on n vertces and m edges, the quantty m n + 1 s called the cyclomatc number of G. G s called a uncyclc graph f ts cyclomatc number s 1. Theorem 3.. Let G be a connected DLS graph on n vertces wth cyclomatc number c n 5, and G s connected. Let H be a graph that s L-cospectral wth G K r. Then one of the followng holds: (a) H s somorphc to G K r ; (b) H = N K 1 K r 1, where N s a graph on n 1 vertces and c + 1 edges. In ths case, n s a Laplacan egenvalue of G, the algebrac connectvty of G s 1, and G has 1 as a Laplacan egenvalue wth multplcty at least. Proof. Lemma.1 mples that H and G rk 1 are L-cospectral. Lemma.8 mples that H has r + 1 components. Suppose that H = H 0 H 1 H r, where H s a connected graph on n vertces and m edges ( = 0, 1,..., r). Wthout loss of generalty, assume that n 0 n 1 n r 1. Snce the cyclomatc number of G s c, G has n vertces and c + n 1 edges. So G has n vertces and (n 1)(n ) c edges. By Lemma.8 we have (n 1)(n ) n = n + r, m = c. By n r 1, we have n 0 n. So we can consder the followng three cases. Case 1. If n 0 = n, then n 1 = n = = n r = 1. Hence H = H 0 rk 1. Snce H and G rk 1 are L-cospectral, G and H 0 are L-cospectral. Snce G s DLS, by Lemma.4, G s also DLS. Hence H 0 = G, H = G rk 1. In ths case, H s somorphc to G K r,.e., part (a) holds. Case. If n 0 = n 1, then n 1 =, n = n 3 = = n r = 1. Hence H 0 has n 1 vertces, H 1 = K, H = H 3 = = H r = K 1. Snce H = H 0 H 1 H r and G rk 1 are L-cospectral, H 0 K and G K 1 are L-cospectral. By Lemma.8 we have m = (n 1)(n ) c, m 0 = (n 1)(n ) (c + 1). Hence H 0 has n 1 vertces and c + 1 edges and H = H 0 K 1 K r 1. Snce H 0 K s L-cospectral wth G K 1, H 0 and G have the same L-ndex, and s a Laplacan egenvalue of G ( s a Laplacan egenvalue of K ). Lemma.1 mples that n s a Laplacan egenvalue of G. Note that H 0 has n 1 vertces and c + 1 edges. Snce c n 5, H 0 has at least 3 components,.e., H 0 has 0 as a Laplacan egenvalue wth multplcty at least 3. Lemma.1 mples that the L-ndex of H 0 s n 1, and ts multplcty s at least. Snce H 0 K and G K 1 are L-cospectral, by Lemma.1, the algebrac connectvty of G s 1, and ts multplcty s at least. Hence part (b) holds.

4 1594 J. Zhou, C. Bu / Dscrete Mathematcs 31 (01) Case 3. Suppose n 0 n. Notce that (n 1)(n ) c = r m r Lemma.9 mples that (n 1)(n ) c = m n (n 1). By 0 c n 5, we have n 5. n (n 1) (n )(n 3) + 3. (1) Snce c n 5, we have (n 1)(n ) c (n )(n 3) + 3. Inequalty (1) mples that (n 1)(n ) (n )(n 3) c = + 3, c = n 5. () By nequalty (1) and Lemma.9, we have n 0 = n, n 1 = 3, n = n 3 = = n r = 1, and H 0 and H 1 are complete graphs. Snce H = H 0 H 1 H r and G rk 1 are L-cospectral, K n K 3 and G K 1 are L-cospectral. If n > 7, by Lemma.13, K n K 3 and G K 1 are somorphc, a contradcton. So we have 5 n 7. If n = 5, the Laplacan spectra of K n K 3 and G K 1 are both 3, 3, 3, 3, 0, 0. Lemma.14 mples that the number of spannng trees of G s 81 5, a contradcton. If n = 6, the Laplacan spectrum of G s 4, 4, 4, 3, 3, 0. Lemma.1 mples that the Laplacan spectrum of G s 3, 3,,,, 0. By Lemma.14, the number of spannng trees of G s 1. From Eq. () we have c = n 5 = 1. Hence G s a uncyclc graph on 6 vertces, the number of spannng trees of G s smaller than or equal to 6, a contradcton. If n = 7, the Laplacan spectrum of G s 5, 5, 5, 5, 3, 3, 0. Lemma.14 mples that the number of spannng trees of G s 9 5 4, a contradcton. 7 For a connected graph G on n vertces, Lemma. mples that G s dsconnected f and only f the L-ndex of G s n. Snce the L-ndex of G s n f and only f G s the product of two graphs (cf. [4, Lemma.7]), G s connected f and only f G s not the product of two graphs. Clearly a DLS tree T has cyclomatc number 0, and T s connected f and only f T s not a star. A DLS uncyclc graph U has cyclomatc number 1, and U s connected f and only f U C 4 or (K (n 3)K 1 ) K 1 (n 3). Note that almost all known connected DLS graphs are trees or uncyclc graphs (see [15,1,13,18,8,3,11]). So most known connected DLS graphs satsfy the condtons gven n Theorem 3.. Let G and H be two L-cospectral graphs. We say that H s a cospectral mate of G, f H s not somorphc to G. Obvously a graph G s DLS f and only f G has no cospectral mates. Theorem 3.3. Let G be a connected DLS graph on n vertces wth cyclomatc number c n 5, and G s connected. If G K 1 s DLS, then G K r s DLS for any postve nteger r. Proof. Assume that G K r has a cospectral mate H. By Theorem 3. we have H = N K 1 K r 1, where N s a graph on n 1 vertces. Lemma.3 mples that G K 1 and N K 1 are L-cospectral. Snce G K 1 s DLS, we know that N K 1 s somorphc to G K 1. So N K s somorphc to G K 1. By G s connected we have G = K, a contradcton to G s connected. Hence G K r has no cospectral mates,.e., G K r s DLS. Theorem 3.4. Let G be a connected DLS graph on n vertces wth cyclomatc number c n 5, G s connected, the maxmum degree of G s smaller than n. Then G K r s DLS for any postve nteger r. Proof. If G K r has a cospectral mate, by Theorem 3., we know that n s a Laplacan egenvalue of G. Let µ(g) be the L-ndex of G, then µ(g) n. Snce the maxmum degree of G s smaller than n, by Lemma.7, we have µ 1 < n, a contradcton. Hence G K r has no cospectral mates,.e., G K r s DLS. Theorem 3.5. Let G be a connected DLS graph on n vertces wth cyclomatc number c n 5, G s connected, the vertex connectvty κ(g) = 1. Then G K r s DLS for any postve nteger r. Proof. Snce G s connected, G s not a complete graph. Let a(g) be the algebrac connectvty of G. If G K r has a cospectral mate, by Theorem 3., we have a(g) = 1. Snce κ(g) = 1, by Lemma.6, G has a vertex v such that v s adjacent to every other vertex of G. In ths case, G s dsconnected, a contradcton to G s connected. Hence G K r has no cospectral mates,.e., G K r s DLS. An -graph, denoted by G s,t, s a graph consstng of cycles C s and C t wth just one vertex n common (see Fg. 1). Clearly an -graph has cyclomatc number. If G s,t has no trangles, then t has at least 7 vertces and ts complement s connected. It s known that an -graph G s,t wthout trangles s DLS (cf. [1, Theorem 5.1]). Theorem 3.5 mples that G s,t K r s DLS f G s,t has no trangles. Corollary 3.6. Let G be a DLS tree on n vertces and n 5. Then G K r s DLS for any postve nteger r. Proof. If G s dsconnected, then G = K 1,n 1. By Lemma.10, G K r s DLS. If G s connected, by Theorem 3.5, G K r s DLS.

5 J. Zhou, C. Bu / Dscrete Mathematcs 31 (01) Fg. 1. The -graph G s,t. Some DLS trees are gven n [15,1,13,18]. Corollary 3.7. Let G be a DLS uncyclc graph on n vertces and n 6. Then G K r s DLS when G s not a cycle of order 6. Proof. If G s dsconnected, by n 6, we have G = (K (n 3)K 1 ) K 1. By Lemma.11, G K r s DLS. So we only need to consder the case that G s connected. If G s a cycle, by Lemma.1, G K r s DLS. If G s not a cycle, then the vertex connectvty of G s 1. By Theorem 3.5, G K r s DLS. Some DLS uncyclc graphs can be found n [8,3,11]. 4. Some observatons Let G be a connected DLS graph on n vertces, and G satsfes the condtons gven n Theorem 3.. If G K r has a cospectral mate, by Theorems 3. and 3.5, the followng facts hold. (1) n s a Laplacan egenvalue of G. () The algebrac connectvty of G s 1, and ts multplcty s at least. (3) G s a -connected graph. (If G has a cut vertex, by Theorem 3.5, G s DLS.) (4) Let µ(g) be the L-ndex of G, then n µ(g) < n. (Snce G s connected, by Lemma., we have µ(g) < n.) Most known connected DLS graphs do not satsfy the above four facts smultaneously (most known connected DLS graphs have cut vertces). In [4], Zhang et al. showed that wheel graph C 6 K 1 has a cospectral mate (K K 1 ) K 1. Cycle C 6 s a DLS graph satsfyng the condtons of Theorem 3.. Graph K K 1 has 5 vertces and edges. The Laplacan egenvalues of C 6 are 4, 3, 3, 1, 1, 0. Clearly C 6 K r and (K K 1 ) K 1 K r 1 satsfy the condtons of part (b) of Theorem 3.. Acknowledgments The authors would lke to thank the Edtor and two revewers for ther careful readng and valuable comments. References [1] R. Boulet, The centpede s determned by ts Laplacan spectrum, C. R. Acad. Sc. Pars, Ser. I 346 (008) [] R. Boulet, Dsjont unons of complete graphs characterzed by ther Laplacan spectrum, Electron. J. Lnear Algebra 18 (009) [3] R. Boulet, Spectral characterzatons of sun graphs and broken sun graphs, Dscrete Math. Theor. Comput. Sc. 11 (009) [4] R. Boulet, B. Jouve, The lollpop graphs s determned by ts spectrum, Electron. J. Combn. 15 (008) #R74. [5] C. Bu, J. Zhou, Starlke trees whose maxmum degree exceed 4 are determned by ther Q -spectra, Lnear Algebra Appl. 436 (01) [6] D. Cvetkovć, P. Rowlnson, S. Smć, An Introducton to the Theory of Graph Spectra, Cambrdge Unversty Press, Cambrdge, 010. [7] D. Cvetkovć, S. Smć, Z. Stanć, Spectral determnaton of graphs whose components are paths and cycles, Comput. Math. Appl. 59 (010) [8] W.H. Haemers, X. Lu, Y. Zhang, Spectral characterzatons of lollpop graphs, Lnear Algebra Appl. 48 (008) [9] S.J. Krkland, J.J. Molterno, M. Neumann, B. Shader, On graphs wth equal algebrac and vertex connectvty, Lnear Algebra Appl. 341 (00) [10] Y. Ln, J. Shu, Y. Meng, Laplacan spectrum characterzaton of extensons of vertces of wheel graphs and mult-fan graphs, Comput. Math. Appl. 60 (010) [11] X. Lu, S. Wang, Y. Zhang, X. Yong, On the spectral characterzaton of some uncyclc graphs, Dscrete Math. 311 (011) [1] X. Lu, Y. Zhang, X. Gu, The mult-fan graphs are determned by ther Laplacan spectra, Dscrete Math. 308 (008) [13] X. Lu, Y. Zhang, P. Lu, One specal double starlke graph s determned by ts Laplacan spectrum, Appl. Math. Lett. (009) [14] H. Ma, H. Ren, On the spectral characterzaton of the unon of complete multpartte graph and some solated vertces, Dscrete Math. 310 (010) [15] G.R. Omd, K. Tajbakhsh, Starlke trees are determned by ther Laplacan spectrum, Lnear Algebra Appl. 4 (007) [16] X. Shen, Y. Hou, Y. Zhang, Graph Z n and some graphs related to Z n are determned by ther spectrum, Lnear Algebra Appl. 404 (005) [17] S. Smć, Z. Stanć, On some forests determned by ther Laplacan or sgnless Laplacan spectrum, Comput. Math. Appl. 58 (009) [18] Z. Stanć, On determnaton of caterpllars wth four termnal vertces by ther Laplacan spectrum, Lnear Algebra Appl. 431 (009) [19] E.R. van Dam, W.H. Haemers, Whch graphs are determned by ther spectra? Lnear Algebra Appl. 373 (003) [0] E.R. van Dam, W.H. Haemers, Developments on spectral characterzatons of graphs, Dscrete Math. 309 (009) [1] J.F. Wang, Q.X. Huang, F. Belardo, E.M. L Marz, On the spectral characterzatons of -graphs, Dscrete Math. 310 (010) [] J.F. Wang, S. Smć, Q.X. Huang, F. Belardo, E.M. L Marz, Laplacan spectral characterzaton of dsjont unon of paths and cycles, Lnear Multlnear Algebra 59 (011) [3] X.D. Zhang, Two sharp upper bounds for the Laplacan egenvalues, Lnear Algebra Appl. 376 (004) [4] Y. Zhang, X. Lu, X. Yong, Whch wheel graphs are determned by ther Laplacan spectra? Comput. Math. Appl. 58 (009) [5] Y. Zhang, X. Lu, B. Zhang, X. Yong, The lollpop graph s determned by ts Q -spectrum, Dscrete Math. 309 (009)