The (signless) Laplacian spectral radii of c-cyclic graphs with n vertices, girth g and k pendant vertices

Transcription

1 Linear and Multilinear Algebra ISSN: Print Online Journal homepage: The signless Laplacian spectral radii of c-cyclic graphs with n vertices, girth g and k pendant vertices Muhuo Liu, Kinkar Ch. Das & Hong-Jian Lai To cite this article: Muhuo Liu, Kinkar Ch. Das & Hong-Jian Lai 017 The signless Laplacian spectral radii of c-cyclic graphs with n vertices, girth g and k pendant vertices, Linear and Multilinear Algebra, 65:5, , DOI: / To link to this article: Published online: 01 Aug 016. Submit your article to this journal Article views: 68 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at Download by: [West Virginia University Libraries] Date: 7 March 017, At: 07:0

2 LINEAR AND MULTILINEAR ALGEBRA, 017 VOL. 65, NO. 5, The signless Laplacian spectral radii of c-cyclic graphs with n vertices, girth g and k pendant vertices Muhuo Liu a, Kinkar Ch. Das b, Hong-Jian Lai c a Department of Mathematics, South China Agricultural University, Guangzhou, China; b Department of Mathematics, Sungkyunkwan University, Suwon, Republic of Korea; c Department of Mathematics, West Virginia University, Morgantown, WV, USA ABSTRACT Let Ɣ g n, k; c denote the class of c-cyclic graphs with n vertices, girth g 3andk 1 pendant vertices. In this paper, we determine the unique extremal graph with largest signless Laplacian spectral radius and Laplacian spectral radius in the class of connected c- cyclic graphs with n cg vertices, girth g and at most n cg 1 1 pendant vertices, respectively, and the unique extremal graph with largest signless Laplacian spectral radius of Ɣ g n, k; c when n cg 1 + k + 1andc 1, and we also identify the unique extremal graph with largest Laplacian spectral radius in Ɣ g n, k; c in the case c 1 and either n cg 1 + k + 1andg is even or n 1 g 1k+cg and g is odd. Our results extends the corresponding results of [Sci. Sin. Math. 010;40: , Electron. J. Combin. 011; 18:p.183, Comput. Math. Appl. 010;59: , Electron. J. Linear Algebra. 011;: and J. Math. Res. Appl. 014;34: ]. ARTICLE HISTORY Received 3 March 016 Accepted 5 July 016 COMMUNICATED BY J.Y. Shao KEYWORDS c-cyclic graph; girth; pendant vertices; adjacency spectral radius; signless Laplacian spectral radius AMS SUBJECT CLASSIFICATIONS 05C50; 05C35; 05C75 1. Introduction Throughout this paper, unless specially indicated, we are concerned with connected undirected simple graph only. Suppose G isagraphwithvertexsetvg ={v 1, v,..., v n }. Foravertexv of G, weusen G v and d G v to denote the neighbour set and degree of v in G, respectively. If there is no confusion, we always simply write du and Nu instead of d G u and N G u, respectively. The sequence π = d 1, d,..., d n is called the degree sequence of G if d i = dv i holds for 1 i n. Throughout this paper, we enumerate the degrees in non-increasing order, i.e. d 1 d d n, and we suppose that dv i = d i, where 1 i n.especially,weuse G to denote the maximum degree of G.Fromthe definition, it follows that G = d 1 G. Wecallu a pendant vertex if du = 1, and call u a maximum degree vertex if du = G. Suppose that P is a path. If one end vertex of P is a pendant vertex while all the internal vertices of P are vertices with degrees two, then P is called a pendant path. Throughout this paper, k and c are two nonnegative integers, and n is a positive integer. If G is connected with n vertices and n + c 1 edges, then G is called a c-cyclic graph.in particular, G is called a tree, unicyclic graph, bicyclic graph or a tricyclic graph if c = 0, CONTACT Hong-Jian Lai hjlai@math.wvu.edu 016 Informa UK Limited, trading as Taylor & Francis Group

3 870 M. LIU ET AL. 1, or 3, respectively. The length of a shortest cycle of G is called the girth of G and denoted by gg. LetƔn, k; c denote the class of c-cyclic graphs with n vertices and k pendant vertices, let Ɣ g n; c denote the class of c-cyclic graphs with n vertices and girth g, andletɣ g n, k; c denote the class of c-cyclic graphs with n vertices, k pendant vertices and girth g, whereg is an integer being at least three hereafter. It is easy to see that Ɣn, k; c = n g=3 Ɣ g n, k; c and Ɣ g n; c = n 1 k=0 Ɣ gn, k; c. For simplification, let Tn, k, Un, k, Bn, k and Sn, k be the class of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs with n vertices and k pendant vertices, respectively. As usual, K n, C n, P n and K s,n s define, respectively, the complete graph, cycle, path and complete bipartite graph on n vertices. Suppose v is a vertex of G, andp s = w 1 w w s, where VP s VG =. If we obtain a new graph G from G and P s by adding two edges vw 1 and vw s,thenwesaythatg is obtained from G by sewing the path P s to v of G. If we obtain a new graph G from G and P s by adding one edge vw 1,thenwesaythatG is obtained from G by attaching the path P s to v of G. In the sequel, if we say that we attach or sew k pathsto one vertexof G, then we agree that these k paths are vertex disjoint each other, and they are also vertex disjoint with G. If q is a positive integer and G is a connected graph, qg denote the graph consisting of q copies of the graph G,andq p means p copies of the integer q,wherepis a nonnegative integer. Paths P l1, P l,..., P lk are said to have almost equal lengths if l 1, l,..., l k satisfy l i l j 1for1 i j k. Denoted by Cq 1, q,..., q c, the graph on c i=1 q i vertices obtained by sewing the paths P q1 1, P q 1,..., P qc 1, to a common vertex, where q i and1 i c. LetF n k, C q s 1, C c s be the c-cyclic graph on n vertices obtained from Cq s 1, qc s by attaching k paths of almost equal lengths to the maximum degree vertex of Cq s 1, qc s. In particular, we always simply write F n k, C q c 1, C q 0 and F n k, C q 0 1, C q c as F n k, C q c 1 and F n k, C q c, respectively. Furthermore, we use the symbol F n k = F n k, C g 0 to denote the unique tree on n vertices obtained by attaching k paths of almost equal lengths to a common vertex. If all cycles of G have exactly one common vertex, then G is called a bundle graph see, e.g. [1]. From the definitions, both Cq 1, q,..., q c and F n k, C q s 1, C q c s are bundle graphs. Let DG be the diagonal matrix of vertex degrees, and AG be the adjacency matrix of G. TheLaplacian matrix and signless Laplacian matrix of G are, respectively, defined as LG = DG AG and QG = DG + AG. The maximum eigenvalues of LG and QG are denoted by λg and μg, respectively. Furthermore, μg and λg are called, respectively, the signless Laplacian spectral radius and Laplacian spectral radius of G. For the relation between μg and λg, it is well known that Theorem 1.1 []: If G is a connected graph on n vertices, then q G + 1 λg μg, where the first equality holds if and only if G = n 1, and the second equality holds if and only if G is bipartite. If G has the largest signless Laplacian spectral radius in some given category of graphs, then we call G asignless Laplacian largest extremal graph.

4 LINEAR AND MULTILINEAR ALGEBRA 871 Recently, the work on determining the signless Laplacian largest extremal graph, and/or Laplacian largest extremal graph in Ɣn, k; c, has attained much attention. For any fixed positive number n and k, itisprovedthat:f n k is the unique Laplacian largest extremal tree also the signless Laplacian largest extremal tree by Theorem 1.1 oftn, k,[3,4] F n k, C 1 3 is the unique signless Laplacian largest extremal unicyclic graph of Un, k [5,6] when n k + 3andF n k, C 1 4 is the unique Laplacian largest extremal unicyclic graph of Un, k [5,7]whenn k + 4; F n k, C 3 is the unique signless Laplacian largest extremal bicyclic graph of Bn, k [8 10] whenn k + 5andF n k, C 4 is the unique Laplacian largest extremal bicyclic graph of Bn, k [5,7] whenn k + 7; F n k, C 3 3 is the unique signless Laplacian largest extremal tricyclic graph of Sn, k [3,9,11]whenn k + 7and F n k, C 3 is the unique Laplacian largest extremal tricyclic graph of Sn, k [1] when 4 n k In the sequel, one of the present authors extended the above referred results of [3 1] by determining the unique signless Laplacian and Laplacian largest extremal graphs of Ɣn, k; c for c 0, k 1andn c + k + 1, namely, he proved that Theorem 1. [13]: If k 1, c 0 and n c + k + 1, then F n k, C c 3 is the unique signless Laplacian largest extremal graph of Ɣn, k; c. Theorem 1.3: [13] Suppose that k 1, c 0 and G is a Laplacian largest extremal graph of Ɣn, k; c. i If n 3c + k + 1, then G = F n k, C c 4. ii If n = c + k t and 0 t c 1, then G = F n k, C t 4, Cc t 3. iii If k + 1 n c + k, then G is any graph with G = n 1. At the same time, the extremal graphs with largest signless Laplacian spectral radii in the class of Ɣ g n, k; c and/or Ɣ g n; c were also studied by some scholars. Up to now, for any fixed positive number n, k and g 3, the following results are identified: F n k, C g 1 is the unique Laplacian largest extremal unicyclic graph of Ɣ g n, k; 1 for any k 1and n k + g [14]; F n n g, C g 1 is the unique signless Laplacian largest extremal unicyclic graph of Ɣ g n; 1 for any n g [15]; F n n g + 1, C g is the unique signless Laplacian and Laplacian largest extremal bicyclic graph in the class of bicyclic bundle graphs with n vertices and girth g for any n g 1, respectively, [15 17]; F n n 3g +, C g 3 is the unique signless Laplacian largest extremal tricyclic graph in the class of tricyclic bundle graphs with n vertices and girth g for any n 3g.[18] In this paper, we will extend the corresponding results of [15 18] by showing the following theorem: Theorem 1.4: Let G be the class of graphs pertaining to Ɣ g n; c, which contain at most n cg 1 1 pendant vertices. If g 3, c 1 and n max{cg 1 + 1, 6}, then F n n cg 1 1, C g c is the unique signless Laplacian and Laplacian largest extremal graph of G, respectively. Remark 1.1: Since λk,3 = λf 5 0, C 3 = 5, the condition n 6 in Theorem 1.4 is necessary. Let H 1 be the bicyclic graph with six vertices obtained from K 4 e by attaching two isolated vertices to one vertex of degree three of K 4 e.sinceλf 6 1, C 3 = λh 1 = 6, the condition contain at most n cg 1 1 pendant vertices in Theorem 1.4 is also necessary. We will also extend the corresponding result of [14] by proving the following theorem:

5 87 M. LIU ET AL. Theorem 1.5: Let c 1, g 3 and k 1, and let G be the Laplacian largest extremal graph of Ɣ g n, k; c. i If g is even and n cg 1 + k + 1, then G = F n k, C g c. ii If gisoddandn 1 g 1k + cg, then G = F n k, C g 1, C c 1. Remark 1.: Since λf 13, C 1 5, C1 6 < < 7.19 < λf 13, C 5, the condition n 1 g 1k + cg intheorem1.5 ii is necessary. One can also easily see that Theorem 1.5 extends partially result of Theorem 1.3. Furthermore, the following theorem extends partially result of Theorem 1.. Theorem 1.6: If k 1, g 3, c 1 and n cg 1 + k + 1, thenf n k, C g c is the unique signless Laplacian largest extremal graph of Ɣ g n, k; c.. The proof of Theorem 1.6 Let uv be an edge of G and v be a vertex of G.Letmv denote the average of the degrees of the vertices being adjacent to v, i.e. mv = du/dv. Denote by u Nv uv = dudu + mu + dvdv + mv. du + dv Theorem 1.1 presents a well-known lower bound to λg, while the following result gives a famous upper bound for μg. Lemma.1 [19]: Let G be a connected graph with at least three vertices, and let s = u 0 v 0 = max{ uv : uv EG} and t = max{ uv : uv EG \{u 0 v 0 }}. Then μg + s t with equality holding if and only if G is a regular graph or a bipartite semiregular graph or a path with four vertices. When G is connected, by the Perron Frobenius Theorem of non-negative matrices see e.g. [0], it follows that Lemma.: If G is connected and G G, then μg < μg and λg λg. Furthermore, by the Perron Frobenius Theorem of non-negative matrices, there also exists a unique positive unit eigenvector corresponding to μg. In the sequel, we use f = f v 1, f v,..., f v n T to indicate the unique positive unit eigenvector corresponding to μg,andcallf theperron vector ofg. As we will see later, the following three operations will play an important role in our proofs. Let G uv be the graph obtained from G by deleting the edge uv EG, and let G + uv be the graph obtained from G by adding an edge uv EG. Similarly, G v denoted the graph obtained from G by deleting the vertex v VG. Lemma.3 [4]: Suppose that u, v are two vertices of a connected graph G, and w 1, w,..., w k 1 k dv are some vertices of Nv \ Nu {u}. LetG = G + w 1 u + w u + +w k u w 1 v w v w k v. If f is the Perron vector of G with f u f v, then μg > μg.

6 LINEAR AND MULTILINEAR ALGEBRA 873 Let G u,v define a new graph obtained from G by subdividing the edge uv, i.e. adding a new vertex w and two edges wu, wv in G uv, whereuv EG. Aninternal path, say P = w 1 w w s s, is a path joining w 1 and w s which need not be distinct such that the degrees of w 1 and w s are greater than, while all other vertices if exist w, w 3,..., w s 1 are of degree. Lemma.4 [5]: If G is a connected graph and uv is an edge in an internal path of G, then μg > μg u,v. Suppose v is a vertex of a connected graph G with at least two vertices. Let G s,t t s 1 be the graph obtained from G by attaching two new paths P s = w 1 w w s and P t = u 1 u u t, respectively, to v of G.LetG s 1,t+1 = G s,t w s 1 w s + u t w s. Lemma.5 [5]: Let G be a connected graph with at least two vertices. If t s 1, then μg s,t > μg s 1,t+1. To prove our results, we need to extend Lemma 3.1 of [13] as follows. Lemma.6: Let G be a graph of Ɣn, k; c and G {K,3, C n }, where c 1 and k 0.If λg k + c + 1 or μg k + c + 1, then G is one of the following candidates: i G is obtained by attaching k paths and then sewing another c paths, respectively, to a common vertex; ii G is obtained by adding an edge joining one vertex of a cycle to one vertex of degree two of a path; iii G is obtained by attaching one path to each of two adjacent vertices of a cycle, respectively; iv G is obtained by adding one edge to two nonadjacent vertices of C n ; v G is obtained by adding one edge joining one vertex of C s with one vertex of C n s, where n s s 3. Proof: Suppose the degree sequence of G is d 1, d,..., d n and G {K,3, C n }.Since G Ɣn, k; c,wehave n n + c 1 = d i..1 If d 1 + d k + c + 3, then i=1 n + c 1 = n d i k + c n k + k = n + c 1, a contradiction. i=1 If k 1andd 1 + d k + c + 1, then G is neither regular nor bipartite semiregular. By Theorem 1.1and Lemma.1, λg μg < + d 1 + d = d 1 + d k + c + 1, a contradiction. If k = 0andd 1 + d c, then by Theorem 1.1 and Lemma.1,wehave λg μg d 1 + d c, a contradiction. If k = 0andd 1 + d = c + 1, then by.1, we have d 1 >d d 3 = 3. By Theorem 1.1 and Lemma.1, λg μg d 1 + d c + 1.

7 874 M. LIU ET AL. Furthermore, by Lemma.1, μg = c + 1impliesthatd 1 = c andd = d 3 = =d n = 3, which contradicts.1. Therefore, if λg k + c + 1orμG k + c + 1, then d 1 + d = k + c +. Since G containsexactly k pendant vertices with.1, we have d 1 + d = k + c +, d 3 = d 4 = =d n k = andd n k+1 = =d n = 1.. If d 1 k + c, then by Theorem 1.1 and Lemma.1 with., λg μg + d 1 + d d 1 + d 3 = + k + cd 1 <k+ c + 1, a contradiction. If d 1 = k + c,by.wehaved = d 3 = =d n k = andd n k+1 = d n k+ = =d n = 1, then G is a graph of i.noticethatd 1 = k + c + d k + c by.. Thus, the final possibility is d 1 = k + c 1. By. it follows that d = 3. Next, we shall prove that μg < max { uv : uv EG}..3 If c, since d 1 = k + c 1 3 = d >d 3 = andg = K,3,thenG is neither regular nor bipartite semiregular. In this case,.3 follows from Lemma.1. Now, we consider the case c = 1. Since G = Cn, d n = 1. Thus, G is neither regular nor bipartite semiregular, and hence.3 follows from Lemma.1. Suppose u 0 v 0 = max { uv : uv EG}, wheredu 0 dv 0.SinceG is connected and c 1, du 0 {k + c 1, 3, }. Ifd 1 = k + c 1 = 3 = d,then either c = andk = 0orc = 1andk =. If v 1 v EG, then by Theorem 1.1 and.3 we have λg μg < u 0 v 0 5, a contradiction. Thus, v 1 v EG. Ifc = and k = 0, then G is one graph of iv or v.ifc = 1andk =, then G is one graph of ii or iii. In the following, we only need to consider the case d 1 = k + c 1 4: Case 1 du 0 = d 1 = k + c 1 4. If dv 0 = 3, then du 0 = d 1 and dv 0 = d by.. Therefore, If dv 0 =, then u 0 v 0 d 1 + d + d d + d 1 + d 1 d 1 + d = k + c 1 + k + c k + + c u 0 v 0 d d d d 1 + = k + c + 6 k + c + 1. k + c + 1 If dv 0 = 1, then u 0 v 0 d d d 1 d = k + c <k+ c + 1. k + c

8 LINEAR AND MULTILINEAR ALGEBRA 875 Case du 0 = 3. By., we have d 1 4anddu 0 = 3 = d >d 3, which implies that 1 dv 0. If dv 0 =, then u 0 v 0 d + d 1 + d d 1 + d d + = k + c <k+ c + 1. If dv 0 = 1, then u 0 v 0 d + d d d + 1 = k + c <k+ c + 1. Case 3 du 0 =. Then, 1 dv 0. If dv 0 =, then If dv 0 = 1, then u 0 v d = k + c + 5 <k+ c + 1. u 0 v d = k + c <k+ c + 1. Now, from Theorem 1.1 with.3 and the above discussion, we can conclude that λg μg < u 0 v 0 k + c + 1, a contradiction. Lemma.7: If G is one graph of ii or iii as defined in Lemma.6 and gg = g, then μg < μf n, C g 1. Proof: Let f be the Perron vector of G. We first suppose that G is a graph of iii.without loss of generality, we may suppose that f v 1 f v.letube a vertex of N G v \ VC g. Let G = G + v 1 u v u. By Lemma.3, wehaveμg < μg. Since G is obtained from a cycle C g by attaching two paths to exactly one vertex of C g, by Lemma.5 we have μg μf n, C g 1. Therefore, μg < μf n, C g 1 holds. We secondly suppose that G is a graph of ii. Suppose that v 1 VC g.then,v VC g.iffv 1 f v,sinced G v = 3, it can be proved similarly with the former case. If f v 1 <fv, choose u as a vertex of N G v 1 VC g.letg = G+v u v 1 u. By Lemma.3,wehaveμG < μg. Suppose that N G u ={v, v}.letg 1 = G + vv uv uv, G = G 1 u and G 3 = G 1 + xu,wherex is a pendant vertex of G 1.Sinced G v = 4, vv lies in an internal path of G. By Lemma.4, μg < μg. Furthermore, since G 1 G 3, by Lemma. we have μg = μg 1 < μg 3, and hence μg < μg 3.Notethat G 3 is obtained by attaching two paths to exactly one vertex of C g. By Lemma.5,wehave μg 3 μf n, C g 1. Now, we can conclude that μg < μf n, C g 1 holds. Proof of Theorem 1.6: Suppose that G is a signless Laplacian largest extremal graph of Ɣ g n, k; c. SinceF n k, C g c Ɣ g n, k; c, by the choice of G and Theorem 1.1 it follows that μg μf n k, C g c >k+ c + 1. By Lemmas.5.7, we can conclude that G is obtained by attaching k paths of almost equal lengths to the maximum degree vertex of Cq 1, q,..., q c,whereq 1 q q c 1 q c = g. If q 1 = g, thenq 1 = q = = q c = g and hence G = F n k, C g c, the result already holds. Thus, we only need to consider the case of q 1 g + 1.

9 876 M. LIU ET AL. Suppose that {u, v, w} VC q1 \{v 1 } such that uv EC q1 and vw EC q1.letx be a pendant vertex of G. LetG 1 = G + uw uv vw, G = G 1 v and G 3 = G 1 + xv. Then, G 3 Ɣ g n, k; c. Since d G v 1 = k + c 3, uw lies in an internal path of G.By Lemma.4,wehaveμG < μg.notethatg 1 G 3. By Lemma., μg < μg = μg 1 < μg 3, contradicting the choice of G.Thus,G = F n k, C g c. 3. The proofs of Theorems By Lemma., if we add some edges to a connected graph, the signless Laplacian spectral radius will increase strictly. However, the following result shows that additional edges to a connected graph can result for unchanged Laplacian spectral radius: Lemma 3.1 [1,]: Let v be a vertex of a connected graph G with at least two vertices and let G be obtained from G by attaching k paths of equal lengths to v. If G is obtained fromgbyaddinganys1 s kk 1 edges among these pendant vertices of G, which belong to the referred k paths, then λg = λg. Lemma 3.: If k 1, c s 1 and n cg 1 + k + s + 1, then μf n k, C s, Cc s g < μf n k, C s 1 g., Cc s+1 Proof: Suppose v 1 is the maximum degree vertex of F n k, C s, Cc s g, and suppose that {u, v, w} VC \{v 1 } such that uv EC and vw EC in F n k, C s, Cc s g. Let x be a pendant vertex of F n k, C s, Cc s g.letg 1 = F n k, C s, Cc s g + uw uv vw, G = G 1 v and G 3 = G 1 + xv. Since d G v 1 = k + c 3, uw is contained in an internal path of G. By Lemma.4,wehaveμF n k, C s, Cc s g < μg.notethat G 1 G 3. By Lemma., μg = μg 1 < μg 3. Furthermore, since G 3 is obtained by attaching k paths to the maximum degree vertex of Cg + 1 s 1, g c s+1, by Lemma.5 we have μg 3 μf n k, C s 1, Cc s+1 g. Now, the result follows by combining the above discussion. Suppose that g 3 is an odd number. Let Fn k, Cc s, Cs g be a graph obtained from F n k, C c s, Cs g by deleting every edge, which has the largest distance from the maximum degree vertex of F n k, C c s, Cs g in each cycle C g. From the definition, Fn k, Cc s, Cs g can be also obtained from F n sg 1 k, C c s by attaching s paths of length 1 g 1 1 to the vertex of degree k + c s of F n sg 1 k, C c s. By Lemma 3.1, the following equation holds for any c s 1, k 1andg 3: λfn k, Cc s, Cs g = λf nk, C c s, Cs g. 3.1 Lemma 3.3: Suppose that g is odd, and G is a c-cyclic graph on n vertices obtained by attaching k paths to the vertex with degree cofcg s, q 1, q,..., q c s.ifc s 1,k 1 and q 1 q q c s g + 1, then λg λfn k, Cc s, Cs g = λf nk, C c s, Cs g,

10 LINEAR AND MULTILINEAR ALGEBRA 877 where the first equality holds if and only if G = F n k, C c s, Cs g. Proof: Since v 1 is the maximum degree vertex of G,wehaved G v 1 = k + c.letg 1 be a c s-cyclic graph on n vertices obtained by deleting every edge, which has the largest distance from v 1 in each C g. By Lemma 3.1, λg = λg 1. First we assume that c = s.inthiscase,g 1 is a tree obtained by attaching k + c paths among which at least c paths are P 0.5g 1 toacommonvertex.by3.1withtheorem 1.1 and Lemma.5,wehave λg = λg 1 = μg 1 μf n k, Cc g = λfn k, Cc g. If λg = λfn k, Cc g, thenμg 1 = μfn k, Cc g, and hence G 1 = F n k, C g c by Lemma.5. By the definition of G 1, G = F n k, C g c.conversely,ifg = F n k, C g c,then by 3.1wehaveλG = λfn k, Cc g. So, the result holds for c = s. Next, we assume that c s 1. In this case, gg 1 = q c s g + 1. Case 1 q 1 = g + 1. Now, q 1 = q = = q c s = g + 1andG 1 is a c s-cyclic graph obtained by attaching k + s paths among which at least s paths are P 0.5g 1 to the vertex of degree c s of Cg + 1 c s. By Lemma.5, μg 1 μ Fn, Cs g with equality holding if and only if G 1 = F n k, C c s, Cs g.bytheorem1.1 and Lemma 3.1,wehave λg = λg 1 = μg 1 μ Fn, Cs g = λfn, Cs g. If λg = λfn, Cs g, thenμg 1 = μfn, Cs g, and hence G = F n k, C c s, Cs g. Conversely,ifG = F n k, C c s, Cs g, then by Lemma 3.1 implies that λg = λfn, Cs g. Case q 1 g +. Suppose that {u, v, w} VC q1 \{v 1 } such that uv EC q1 and vw EC q1.letx be a pendent vertex pertaining to a longest pendant path of G. Let G = G 1 + uw uv vw, G 3 = G v and G 4 = G + xv. Since d G3 v 1 = k + c 3, uw is contained in an internal path of G 3. By Lemma.4, wehaveμg 1 < μg 3.Note that G G 4. By Lemma., μg 1 < μg 3 = μg < μg 4. Note that G 4 contains exactly k + s pendant vertices, and at least s pendant vertices are contained in s pendant paths of lengths 1 g 1 initial from the maximum degree vertex of G 4. By repeating the above operation, we can obtain a c s-cyclic graph G 5 such that μg 4 μg 5,whereG 5 is obtained by attaching k + s paths among which at least s paths are P 0.5g 1 to the vertex of degree c s of Cg + 1 c s. By Lemma.5, μg 5 μfn, Cs g. Now, from Theorem 1.1,wehaveλG = λg 1 μg 1 < μg 5 μfn, Cs g = λfn, Cs g.

11 878 M. LIU ET AL. Lemma 3.4: If g is odd, c s 1,k 1 and n 1 g 1k + cg + s, then λ F n k, C c s, Cs g < λ F n k, C c s+1, C g s 1. Proof: Let P be a longest pendant path among these k pendant paths, which are initial from the maximum degree vertex i.e. v 1 off n k, C c s, Cs g,andlety be the pendant vertex of P.If VP 1 g + 1,then n sg 1 + gc s g 1k = 1 g 1k + gc + 1 s, a contradiction. Thus, VP 1 g Let x be a pendant vertex of a pendant path with length 1 g 1, which is initial from v 1 in Fn, Cs g. By the definition of Fn, Cs g, such vertex x must exist. Let G 1 = Fn, Cs g + xy.thengg 1 g + 1andG 1 is a c s + 1-cyclic graph obtained by attaching k + s 1 paths among which at least s 1 paths are P 0.5g 1 to the maximum degree vertex of Cg +1 c s, q,whereq g +1. By Lemmas.4 and.5, μg 1 μfn k, Cc s+1, C g s 1.Sinceg + 1iseven,by3.1, we have Furthermore, since Fn.,wehave μfn k, Cc s+1, C g s 1 = λf n k, Cc s+1, C g s 1. k, C c s, Cs g G 1,by3.1 withtheorem1.1 and Lemma λ F n k, C c s, Cs g = λfn, Cs g = μfn, Cs g < μg 1 λfn k, Cc s+1, C g s 1 = λf n k, C c s+1, C g s 1. Thus, the required inequality holds. Lemma 3.5: If G is one graph of ii or iii as defined in Lemma.6 and gg = g, then λg < λf n, C g 1. Proof: When g is even, by Lemma.7 and Theorem 1.1, the result already holds. Thus, we may suppose that g is odd and n 6 in the following, as n = 5 can be checked easily. If g = 3andG is a graph of iii, it is well-known that [3] λg max{ Nu Nv :uv EG}, and hence λg 5 < λf n, C g 1 by Theorem 1.1.Ifg = 3andG is a graph of ii,let G be the graph obtained from G by deleting one edge with both end vertices are of degrees two in C 3. By Lemmas.1 and 3.1 with Theorem 1.1,wehave λg = λg max{ uv : uv EG }=5 < λfn, C1 g.

12 LINEAR AND MULTILINEAR ALGEBRA 879 If g 5, then by Lemmas.1. it follows that 16 λg μg < < λf7, C1 5 λf n, C1 g. 3. Now, the result follows from 3.1. Proof of Theorem 1.5: When g is even, by Theorems 1.1 and 1.6,wehaveλG μg μf n k, C g c = λf n k, C g c, whereλg = λf n k, C g c holds if and only if G = F n k, C g c.thus,i follows. Now, we turn to prove ii. Since n 1 g 1k + cg + 1, F nk, C g, C c 1 Ɣ gn, k; c. ByTheorem1.1 and the choice of G, λg λf n k, C g, C c 1 k + c + 1. From Lemmas.6 and 3.5, it follows that G is obtained from Cq 1, q,..., q c by attaching k paths to the maximum degree vertex of Cq 1, q,..., q c,whereq 1 q q c 1 q c = g. We suppose that G contains exactly s cycles C g. By Lemma 3.3,wehave λg λ F n k, C c s, Cs g with equality holding if and only if G = F n k, C c s, Cs g.ifs = 1, then the result already holds. Otherwise, s. By Lemma 3.4, it follows that λ F n k, C c s, Cs g < λ F n k, C c s+1, C g s 1 λf n k, C c 1, C1 g. Thus, λg < λf n k, C c 1, C1 g. This completes the proof of ii. Lemma 3.6: Let G be one graph of iv or v as defined in Lemma.6 and gg = g. If n g 1, then λg < λf n n g + 1, C g and μg < μf n n g + 1, C g. Proof: In this case, G is neither regular nor bipartite semiregular. If n g + 1 1, by Lemma.1 and Theorem 1.1,wehave λg μg < 6 λf n n g + 1, C g μf n n g + 1, C g, and the result already holds. If n = g 1, it is easy to check the result holds for g = 3. Now, we suppose that g 4. Since F7, C1 5 F n0, C g,bytheorem1.1, Lemmas.1. and 3., we have λg μg < < λf7, C1 5 λf n0, C g μf n 0, C g. This completes the proof of this result. Proof of Theorem 1.4: Let G be a Laplacian or signless Laplacian largest extremal graph of G. SinceF n k, C g c G, by Lemmas.6.7 and Lemmas , G is obtained from Cq 1, q,..., q c by attaching k paths to the maximum degree vertex of Cq 1, q,..., q c, where q 1 q q c 1 q c = g and 0 k n cg 1 1. Furthermore, when n = cg 1 + 1, then G = F n 0, C g c and the result already holds. We may suppose that n cg 1 + in the following.

13 880 M. LIU ET AL. If k n cg 1, by Lemma.1 and Theorem 1.1,wehave λg μg d 1 + d = c + k + c + n cg 1 λf n n cg 1 1, C g c μf n n cg 1 1, C g c. By the structure of G and Lemma.1, G is regular with d 1 = c + k = = d.thus,c = 1 and k = 0, which implies that G = C n. In this case, 0 k n cg 1 = 1, a contradiction. Therefore, k = n cg 1 1 and hence G = F n n cg 1 1, C g c. Acknowledgements The authors would like to thank the referees for their valuable comments, corrections and suggestions, which lead to an improvement of the original manuscript. Disclosure statement No potential conflict of interest was reported by the authors. Funding The first author is partially supported by NSFC project [grant number ], the Training Program for Outstanding Young Teachers in University of Guangdong Province [grant number YQ01507]; China Scholarship Council, and the second author is partially supported by the National Research Foundation funded by the Korean government with the [grant number 013R1A1A009341]. References [1] Borovićanin B, Petrović M. On the index of cactuses with n vertices. Publ. Inst. Math. Beograd. 006;79: [] Merris R. Laplacian matrices of graphs: a survey. Linear Algebra Appl. 1994; : [3] Li K, Wang L, Zhao G. The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices. Linear Algebra Appl. 011;435: [4] Zhang X-D. The Laplacian spectral radii of trees with degree sequences. Discrete Math. 008;308: [5] Liu M, Tan X, Liu B. The signless Laplacian spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices. Czech. Math. J. 010;60: [6] Zhang X-D. The signless Laplacian spectral radius of graphs with given degree sequences. Discrete Appl. Math. 009;157: [7] Guo J-M. The Laplacian spectral radii of unicyclic and bicyclic graphs with n vertices and k pendant vertices. Sci. China Math. 010;53: [8] Feng L. The signless Laplacian spectral radius for bicyclic graphs with k pendant vertices. Kyungpook Math. J. 010;50: [9] Liu M, Tan X, Liu B. The largest signless Laplacian spectral radius of connected bicyclic and tricyclic graphs with n vertices and k pendant vertices. Appl. Math. J. Chinese Univ. Ser. A. 011;6:15. Chinese. [10] Huang Y, Liu B, Liu Y. The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences. Discrete Math. 011;311: [11] Zhang J, Guo J-M. The signless Laplacian spectral radius of tricyclic graphs with k pendant vertices. J. Math. Res. Appl. 01;3:81 87.

14 LINEAR AND MULTILINEAR ALGEBRA 881 [1] Guo S-G, Wang Y-F. The Laplacian spectral radius of tricyclic graphs with n vertices and k pendant vertices. Linear Algebra Appl. 009;431: [13] Liu M. The signless Laplacian spectral radii of c-cyclic graphs with n vertices and k pendant vertices. Electron. J. Linear Algebra. 01;3: [14] Liu H, Lu M, Zhang S. On the Laplacian spectral radius of unicyclic graphs with girth g and k pendant vertices. Sci. Sin. Math. 010;40: Chinese. [15] Li K, Wang L, Zhao G. The signless Laplacian spectral radius of unicyclic and bicyclic graphs with a given girth. Electron. J. Combin. 011;18:p.183. [16] Zhai M, Yu G, Shu J. The Laplacian spectral radius of bicyclic graphs with a given girth. Comput. Math. Appl. 010;59: [17] Zhu Z. The signless Laplacian spectral radius of bicyclic graphs with a given girth. Electron. J. Linear Algebra. 011;: [18] Lu Q, Wang L. The signless Laplacian spectral radius of tricyclic graphs with a given girth. J. Math. Res. Appl. 014;34: [19] Pan Y-L. Sharp upper bounds for the Laplacian graph eigenvalues. Linear Algebra Appl. 00;355: [0] Horn RA, Johnson CR. Matrix analysis. New York NY: Cambridge University Press; [1] Das KC. The Laplacian spectrum of a graph. Comput. Math. Appl. 004;48: [] Guo J-M. The Laplacian spectral radius of a graph under perturbation. Comput. Appl. Math. 007;54: [3] Das KC. An improved upper bound for Laplacian graph eigenvalues. Linear Algebra Appl. 003;368:69 78.