Linear Algebra and its Applications

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1 Linear Algebra and its Applications 435 (2011) Contents lists aailable at ScienceDirect Linear Algebra and its Applications jornal homepage: On the Estrada and Laplacian Estrada indices of graphs Zhibin D a,, Zhongzh Li b a Department of Mathematics, Tongji Uniersity, Shanghai , China b College of Mathematics and Software Science, Sichan Normal Uniersity, Chengd , China ARTICLE INFO Article history: Receied 15 Febrary 2011 Accepted 28 March 2011 Aailable online 4 May 2011 SbmittedbyR.A.Braldi AMS classification: 05C35 05C50 Keywords: Estrada index Laplacian Estrada index Spectral moments Closed walks Edge grafting operation Tree ABSTRACT The Estrada index of a graph G is defined as EE(G) = n i=1 eλ i,where λ 1,λ 2,...,λ n are the eigenales of G. The Laplacian Estrada index of a graph G is defined as LEE(G) = n i=1 eμ i,whereμ 1,μ 2,...,μ n are the Laplacian eigenales of G. An edge grafting operation on a graph moes a pendent edge between two pendent paths. We stdy the change of Estrada index of graph nder edge grafting operation between two pendent paths at two adjacent ertices. As the application, we gie the reslt on the change of Laplacian Estrada index of bipartite graph nder edge grafting operation between two pendent paths at the same ertex. We also determine the niqe tree with minimm Laplacian Estrada index among the set of trees with gien maximm degree, and the niqe trees with maximm Laplacian Estrada indices among the set of trees with gien diameter, nmber of pendent ertices, matching nmber, independence nmber and domination nmber, respectiely Elseier Inc. All rights resered. 1. Introdction LetG be asimplegraphwithertexsetv(g) andedge sete(g), where V(G) =n. LetA(G) be the adjacency matrix of G.Denotedbyλ 1,λ 2,...,λ n the eigenales of G (i.e., the eigenales of A(G))[3]. Let L(G) = D(G) A(G) be the Laplacian matrix of G, where D(G) is the diagonal matrix of ertex degrees of G.Denotebyμ 1,μ 2,...,μ n the Laplacian eigenales of G (i.e., the eigenales of L(G))[31]. The Estrada index is a newly proposed graph spectrm-based inariant, which is defined as [11] n EE(G) = e λ i. i=1 Corresponding athor. addresses: zhibind@126.com (Z. D), zhongzhli@126.com (Z. Li) /$ - see front matter 2011 Elseier Inc. All rights resered. doi: /j.laa

2 2066 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) It fond arios applications in a large ariety of problems. Estrada index gies maximm ales for the most folded strctres, ths it is sefl in the measre of folding of the moleclar strctres [11], especially protein chain [12,13]. Estrada index is also an effectie method to measre the centrality of complex networks [14,15], extended atomic branching [16] and the carbon-atom skeleton [22]. In the last few years, Estrada index has attracted more and more attentions of mathematicians. A nmber of lower and pper bonds for Estrada index were established. Gtman et al. [21] and de la Peña et al. [17] estimated Estrada index in terms of the nmbers of ertices and edges, and established bonds for Estrada index inoling graph energy. Bamdad et al. [1] gae a lower bond of Estrada index in terms of the nmbers of positie, zero, negatie adjacency eigenales and graph energy. Recently, the trees with extremal Estrada indices were characterized. The Estrada indices of trees are maximized by the star [4,5,17,37]and minimized by the path [5]. Ilić and Steanoić[24]determined the niqe tree with minimm Estrada index among trees with gien maximm degree. Zhang et al. [34] determined the niqe trees with maximm Estrada indices among trees with gien matching nmber. Li [27] determined the niqe tree with maximm Estrada index among trees with gien bipartition.d and Zho [9] determined the niqe trees with maximm Estrada indices among trees with gien nmber of pendent ertices, independence nmber and domination nmber, respectiely. More reslts on Estrada index can be fond in [6,10,18,19,26,35,36]. In fll analogy with Estrada index, Fath-Tabar et al. [18] proposed the Laplacian Estrada index, which is defined as n LEE(G) = e μ i. i=1 Independently of [18], Li et al. [28] proposed the Laplacian Estrada index of another form, which is defined as n LEE Li (G) = e μi 2m/n, i=1 where n and m are, respectiely, the nmbers of ertices and edges of G.In[28,29], Li et al. established bonds for LEE Li (G) in terms of the nmbers of ertices, edges and graph Laplacian energy. Clearly, LEE(G) = e 2m/n LEE Li (G). Ths, the two definitions of Laplacian Estrada index are actally eqialent. In the following, we se the definition of Laplacian Estrada index proposed by Fath-Tabar et al. in [18]. Zho and Gtman [38] established lower and pper bonds of Laplacian Estrada index in terms of the nmbers of ertices, edges and the first Zagreb index, they also obtained a relationship between the Laplacian Estrada index of a bipartite graph and the Estrada index of its line graph. Bamdad et al. [1] gae a lower bond of Laplacian Estrada index in terms of the nmbers of ertices, edges and connected components. Ilić and Zho [25] proed that the Laplacian Estrada indices of trees are maximized by the star and minimized by the path, which showed the se of Laplacian Estrada index as a measre of branching in alkanes. D [8] determined the tree with maximm Laplacian Estrada index among trees with gien bipartition, and characterized the trees with the first six maximm Laplacian Estrada indices. More reslts on Laplacian Estrada index were reported in [7,39,40]. A latest srey on Estrada index and Laplacian Estrada index can be fond in [23]. These reslts prompt s to stdy more properties for the two noel graph inariants. A pendent path at in a graph G is a path in which no ertex other than is incident with any edge of G otside the path, where the degree of is at least three. An edge grafting operation on a graph moes a pendent edge between two pendent paths at two ertices (not necessarily distinct). Taking the graphs G 1, G 2 showninfig.1 as an example, where G 2 is the graph obtained from G 1 by deleting the edge xy and adding the edge zy, wesayg 2 is obtained nder an edge grafting operation on G 1. The edge grafting operation on graph was often considered and sed in the stdy of arios graph inariants, e.g., spectral radis [30], Laplacian spectral radis [20], graph energy [32,33]. Recently, Ilić and Steanoić [24] stdied the change of Estrada index of graph nder edge grafting operation between two pendent paths at the same ertex.

3 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) Fig. 1. An example on edge grafting operation. The paper is organized as follows. In Section 3, a transformation that increases the Estrada index is presented. Motiated by [24], in Section 4 we stdy the change of Estrada index of graph nder edge grafting operation between two pendent paths at two adjacent ertices. As the application, in Section 5 we gie the reslt on the change of Laplacian Estrada index of bipartite graph nder edge grafting operation between two pendent paths at the same ertex. In Sections 6 and 7, we characterize the trees with extremal Laplacian Estrada indices, inclding the niqe tree with minimm Laplacian Estrada index among the set of trees with gien maximm degree, and the niqe trees with maximm Laplacian Estrada indices among the set of trees with gien diameter, nmber of pendent ertices, matching nmber, independence nmber and domination nmber, respectiely. 2. Preliminaries Denote by M k (G) the kth spectral moment of the graph G, i.e., M k (G) = n that M k (G) is eqal to the nmber of closed walks of length k in G (see [3]). Then EE(G) = k=0 M k (G). k! i=1 λk i. It is well-known Let G 1 and G 2 be two graphs. If M k (G 1 ) M k (G 2 ) for all positie integers k,thenee(g 1 ) EE(G 2 ). Moreoer, if M k0 (G 1 )<M k0 (G 2 ) for some positie integer k 0,thenEE(G 1 )<EE(G 2 ). Let, V(G) (not necessarily = ). A walk is said to be a (, )-walk if it starts at and ends at in G. LetW k (G;, ) be the set of (, )-walks of length k in G. LetM k (G;, ) = W k (G;, ). Clearly, M k (G;, ) = M k (G;, ) for all positie integers k (see [3]). Let 1, 1 V(G 1 ) and 2, 2 V(G 2 ).IfM k (G 1 ; 1, 1 ) M k (G 2 ; 2, 2 ) for all positie integers k,thenwewrite(g 1 ; 1, 1 ) (G 2 ; 2, 2 ).If(G 1 ; 1, 1 ) (G 2 ; 2, 2 ) and there is at least one positie integer k 0 sch that M k0 (G 1 ; 1, 1 )<M k0 (G 2 ; 2, 2 ),thenwewrite(g 1 ; 1, 1 ) (G 2 ; 2, 2 ). For conenience, let M k (G; ) = M k (G;, ), and (G 1 ; 1, 1 ) (G 2 ; 2, 2 ) (G 1 ; 1 ) (G 2 ; 2 ), (G 1 ; 1, 1 ) (G 2 ; 2, 2 ) (G 1 ; 1 ) (G 2 ; 2 ). Let P n be the path on n ertices. Let d G () be the degree of in G.Letd G (, ) be the distance from to in a connected graph G. For a sbset M of the edge set of the graph G, G M denotes the graph obtained from G by deleting the edges in M, and for a sbset M of the edge set of the complement of G, G + M denotes the graph obtained from G by adding the edges in M.For V(G) let G be the graph obtained from G by deleting and its incident edges. 3. Lemmas Let H be a graph (not necessarily connected) with, V(H). Sppose that w i V(H), and w i, w i E(H) for 1 i r. LetE ={w 1, w 2,...,w r } and E ={w 1, w 2,...,w r }.Let H = H + E and H = H + E.

4 2068 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) Let V ={, w 1, w 2,...,w r } and V ={, w 1, w 2,...,w r }.Clearly,V \{} =V \{}. For x 1, x 2 V (x 1, x 2 V, respectiely), let T k (H ; x 1, x 2 ) (T k (H ; x 1, x 2 ), respectiely) be the set of (x 1, x 2 )-walks of length k in H (H, respectiely) starting and ending at the edge(s) in E (E, respectiely). Lemma 3.1. Sppose that (H; ) (H; ) and (H; w i, ) (H; w i, ) for 1 i r. Then for any positie integer k, (i) T k (H ;, ) T k (H ;, ) ; (ii) T k (H ;, x 2 ) T k (H ;, x 2 ) for x 2 V \{}; (iii) T k (H ; x 1, ) T k (H ; x 1, ) for x 1 V \{}; (i) T k (H ; x 1, x 2 ) T k (H ; x 1, x 2 ) for x 1, x 2 V \{}. Proof. We only proe (i). The proofs of (ii), (iii) and (i) are similar. Let k be any positie integer. We may decompose any W T k (H ;, ) into fie possible types of sections as follows: (a) a walk in H with all edges in E ; (b) a (, )-walk in H; (c) a (w i, )-walk in H, where 1 i r; (d) a (, w i )-walk in H, where 1 i r; (e) a (w i, w j )-walk in H, where 1 i, j r (not necessarily i = j). On the other hand, we may decompose any W T k (H ;, ) into fie possible types of sections as follows: (a )awalkinh with all edges in E ; (b )a(, )-walk in H; (c )a(w i, )-walk in H, where 1 i r; (d )a(, w i )-walk in H, where 1 i r; (e )a(w i, w j )-walk in H, where 1 i, j r (not necessarily i = j). By replacing by, we may constrct an injection f (a) s edges in E into a walk of length s in H with all edges in E. mapping a walk of length s in H with all Since (H; ) (H; ), we may constrct an injection f (b) s mapping a (, )-walk of length s in H into a (, )-walk of length s in H. Since (H; w i, ) (H; w i, ), we may constrct an injection f (c i) s mapping a (w i, )-walk of length s in H into a (w i, )-walk of length s in H, where 1 i r. Note that M s (H;, w i ) = M s (H; w i, ) and M s (H;, w i ) = M s (H; w i, ) for any positie integer s (see [3]), and ths (H;, w i ) (H;, w i ) for 1 i r. It follows that we may constrct an injection f (d i) s mapping a (, w i )-walk of length s in H into a (, w i )-walk of length s in H, where 1 i r. Finally, we constrct a mapping f from T k (H ;, ) to T k (H ;, ). LetW = W 1 W 2 T k (H ;, ), where W s for s 1 is a walk of length l s of type (a), (b), (c), (d) or (e). Let f (W) = f (W 1 )f (W 2 ), where f (W s ) = f (a) l s (b), f (W s ) = f (c i) (W s ) if W s is of type (d) and (W s ) if W s is of type (a), f (W s ) = f (b) l s (W s ) if W s is of type l s (W s ) if W s is of type (c) and starts at w i, f (W s ) = f (d i) l s ends at w i, and f (W s ) = W s if W s is of type (e). Note that f (W s ) for s 1 is a walk of length l s of type (a ), (b ), (c ), (d )or(e ), and ths f (W) T k (H ;, ) and f is an injection. This implies that T k (H ;, ) T k (H ;, ). The following lemma is a generalization of Lemma 1 in [34] and Lemma 3.2 in [10]. Lemma 3.2. If (H; ) (H; ) and (H; w i, ) (H; w i, ) for 1 i r, then EE(H )<EE(H ).

5 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) Proof. For positie integer k,lets (k) (S (k), respectiely) be the set of closed walks of length k in H (H, respectiely) containing some edges in E (E, respectiely). Then M k (H ) = M k (H) + S (k), M k (H ) = M k (H) + S (k). We need only to show that S (k) S (k) for all positie integers k, and it is strict for some positie integer k 0. We may niqely decompose any W S (k) into three sections, say W 1 W 2 W 3, where W 1 is a walk in H whose length may be zero, W 2 is the longest walk of W in H starting and ending at the edge(s) in E, and W 3 is a walk in H whose length may be zero. By the choice of W 2, we know that W 2 starts at some ertex in V and ends at some ertex in V.Let S (x 1,x 2 ) (k) ={W S (k) : W 2 is an (x 1, x 2 )-walk}, where x 1, x 2 V.Then S (k) = Similarly, let S (x1,x2) x 1,x 2 V + S (x 1,) x 1 V \{} (k) = S (,) (k) + (k) + x 1,x 2 V \{} x 2 V \{} S (x 1,x 2 ) (k) ={W S (k) : W 2 is an (x 1, x 2 )-walk}, where x 1, x 2 V, and ths S (k) = Now we hae Similarly, S (,) (k) = S (x1,x2) x 1,x 2 V + S (x 1,) x 1 V \{} = = S (,) (k) = (k) = S (,) (k) + y V(H) (k) + x 1,x 2 V \{} S (x 1,x 2 ) (k). x 2 V \{} S (x 1,x 2 ) (k). S (,x 2) (k) S (,x 2) (k) M k1 (H; y, ) T k2 (H ;, ) M k3 (H;, y) T k2 (H ;, ) y V(H) T k2 (H ;, ) M k1 +k 3 (H;, ). T k2 (H ;, ) M k1 +k 3 (H;, ). M k1 (H; y, ) M k3 (H;, y) By Lemma 3.1 (i), T s (H ;, ) T s (H ;, ) for all positie integers s. Since(H; ) (H; ), we hae M s (H;, ) M s (H;, ) for all positie integers s, and it is strict for some positie integer s 0. (k) S (,) It follows that S (,) (k), and it is strict for some positie integer k 0.

6 2070 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) Similarly, by Lemma 3.1 (ii), (iii) and (i), we hae S (,x 2) (k) x 2 V \{} = x 2 V \{} T k2 (H ;, x 2 ) M k1 +k 3 (H; x 2, ) x 2 V \{} x 1 V \{} x 1 V \{} x 1,x 2 V \{} x 1,x 2 V \{} S (,x 2) (k) = x 2 V \{} S (x 1,) (k) = x 1 V \{} S (x 1,) (k) = S (x 1,x 2 ) (k) = S (x 1,x 2 ) (k) = x 1 V \{} x 1,x 2 V \{} x 1,x 2 V \{} T k2 (H ;, x 2 ) M k1 +k 3 (H; x 2, ), T k2 (H ; x 1, ) M k1 +k 3 (H;, x 1 ) T k2 (H ; x 1, ) M k1 +k 3 (H;, x 1 ), T k2 (H ; x 1, x 2 ) M k1 +k 3 (H; x 2, x 1 ) T k2 (H ; x 1, x 2 ) M k1 +k 3 (H; x 2, x 1 ). Therefore S (k) S (k) for all positie integers k, and it is strict for some positie integer k The change of Estrada index of graph nder edge grafting operation Let G be a connected graph with at least two ertices, and and be two adjacent ertices in G. For integers a, b with a, b 0, let G (a, b) be the graph obtained from G by attaching two pendent paths P a : x 1 x 2 x a and P b : y 1 y 2 y b at end ertices x 1 and y 1 to, and let G, (a, b) be the graph obtained from G by attaching two pendent paths P a : x 1 x 2 x a and P b : y 1 y 2 y b at end ertices x 1 and y 1, respectiely, to and, seefig.2. ForG (a, b), we reqire that a b 0. Ilić and Steanoić[24] considered the change of Estrada index of graph nder edge grafting operation between two pendent paths at the same ertex, and gae the following reslt. Lemma 4.1 [24]. For integers s, twiths t + 2 2, EE(G (s, t)) < EE(G (s 1, t + 1)). In this section, we consider the change of Estrada index of graph nder edge grafting operation between two pendent paths at two adjacent ertices. Since the Estrada index of a disconnected graph is eqal to the sm of Estrada indices of its connected components, ths we need only to consider the connected graphs. Let s, t be integers with s t If d G () = d G () = 1 (i.e., G = = P 2 ), then both G, (s, t) and G, (t + 1, s 1) are the paths on s + t + 2 ertices, and ths EE(G, (s, t)) = Fig. 2. The graphs G (a, b) and G, (a, b).

7 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) EE(G, (t + 1, s 1)). Ifd G () > 1 and d G () = 1, then G, (s, t) = G, (t + 1, s 1), and ths EE(G, (s, t)) = EE(G, (t + 1, s 1)). Ifd G () = 1 and d G () > 1, then by Lemma 4.1, EE(G, (s, t)) < EE(G, (t + 1, s 1)). Sppose in the following that d G (), d G () >1, and we will show that EE(G, (s, t)) < EE(G, (t + 1, s 1)). Lemma 4.2. Let G = G. Sppose that a, b are two integers with a > b 1. Then (i) (G (a, b); y 1 ) (G (a, b); x 1 ); (ii) (G (a, b); z, y 1 ) (G (a, b); z, x 1 ) for z V(G) \{}. Proof. (i) Let k be any positie integer. For two distinct ertices x, y V(G (a, b)), letw k (G (a, b); x, [y]) be the set of (x, x)-walks of length k in G (a, b) containing y, and let M k (G (a, b); x, [y]) = W k (G (a, b); x, [y]). Note that and M k (G (a, b); y 1 ) = M k (P b ; y 1 ) + M k (G (a, b); y 1, []) M k (G (a, b); x 1 ) = M k (P a ; x 1 ) + M k (G (a, b); x 1, []). Since a > b 1, P b isapropersbgraphofp a, and then (P b ; y 1 ) (P a ; x 1 ).Thsweneedonlyto show that M k (G (a, b); y 1, []) M k (G (a, b); x 1, []). We constrct a mapping f from W k (G (a, b); y 1, []) to W k (G (a, b); x 1, []). For W W k (G (a, b); y 1, []), we may niqely decompose W into three sections, say W 1 W 2 W 3, where W 1 is the shortest (y 1, )-section of W (for which the internal ertices, if exist, are only possible to be y 1, y 2,...,y b ), W 2 is the longest (, )-section of W whose length may be zero, and W 3 is the remaining (, y 1 )-section of W (for which the internal ertices, if exist, are only possible to be y 1, y 2,...,y b ). Let f (W) = f (W 1 )f (W 2 )f (W 3 ), where f (W 1 ) is an (x 1, )-walk obtained from W 1 by replacing y i by x i for i = 1, 2,...,b, f (W 2 ) = W 2, and f (W 3 ) is a (, x 1 )-walk obtained from W 3 by replacing y i by x i for i = 1, 2,...,b. Obiosly,f (W) W k (G (a, b); x 1, []) and f is an injection. Ths M k (G (a, b); y 1, []) M k (G (a, b); x 1, []). (ii) Let z V(G) \{}, and k be any positie integer. We constrct a mapping f from W k (G (a, b); z, y 1 ) to W k (G (a, b); z, x 1 ).ForW W k (G (a, b); z, y 1 ), we may niqely decompose W into two sections, say W 1 W 2, where W 1 is the longest (z, )-section of W, and W 2 is the remaining (, y 1 )- section of W (for which the internal ertices, if exist, are only possible to be y 1, y 2,...,y b ). Let f (W) = f (W 1 )f (W 2 ), where f (W 1 ) = W 1, and f (W 2 ) is a (, x 1 )-walk obtained from W 2 by replacing y i by x i for i = 1, 2,...,b. Obiosly,f (W) W k (G (a, b); z, x 1 ) and f is an injection. Ths M k (G (a, b); z, y 1 ) M k (G (a, b); z, x 1 ). Theorem 4.1. Sppose that G is a connected graph. Let, be two adjacent ertices in G, where d G (), d G () >1. For integers s, twiths t + 2 2, EE(G, (s, t)) < EE(G, (t + 1, s 1)). Proof. Denote by w 1, w 2,...,w r the neighbors of in G different from, where r = d G () 1. Let G = G.Sinces > t+1 1, by Lemma 4.2 (i) and (ii), we hae (G (s, t+1); y 1 ) (G (s, t+1); x 1 ), and (G (s, t + 1); w i, y 1 ) (G (s, t + 1); w i, x 1 ) for 1 i r. Let E y1 ={y 1 w 1, y 1 w 2,...,y 1 w r } and E x1 ={x 1 w 1, x 1 w 2,...,x 1 w r }.Notethat and G, (s, t) = G (s, t + 1) + E y1 G, (t + 1, s 1) = G (s, t + 1) + E x1. Then the reslt follows from Lemma 3.2.

8 2072 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) For x V(G), letn G (x) be the set of neighbors of x in G. If N G () \{} =N G () \{},theng, (a, b) = G, (b, a) for integers a, b with a, b 0, and ths we hae Corollary 4.1. Sppose that G is a connected graph. Let, be two adjacent ertices in G, where d G (), d G () >1. Sppose that N G ()\{} =N G ()\{}. For integers s, twiths t +2 2,EE(G, (s, t)) < EE(G, (t + 1, s 1)) = EE(G, (s 1, t + 1)). 5. The change of Laplacian Estrada index of bipartite graph nder edge grafting operation Let L(G) be the line graph of a graph G. Zho and Gtman [38] gae the following relationship between the Laplacian Estrada index of a bipartite graph and the Estrada index of its line graph. Lemma 5.1 [38]. Let G be a bipartite graph with n ertices and m edges. Then LEE(G) = n m + e 2 EE(L(G)). Now we consider the change of Laplacian Estrada index of bipartite graph nder edge grafting operation between two pendent paths at the same ertex. Theorem 5.1. Let G be a nontriial bipartite connected graph with V(G). For integers s, twith s t + 2 2, LEE(G (s, t)) < LEE(G (s 1, t + 1)). Proof. By Lemma 5.1, we need only to show that EE(L(G (s, t))) < EE(L(G (s 1, t + 1))). Denote by z 1 (z 2, respectiely) the ertex in L(G (1, 1)) corresponding to the edge x 1 (y 1,respectiely) in G (1, 1). Obiosly,z 1 and z 2 are adjacent in L(G (1, 1)), and N L(G (1,1))(z 1 ) \{z 2 }= N L(G (1,1))(z 2 ) \{z 1 }.SinceG is a nontriial connected graph, we hae L(G (1, 1)) is also a connected graph, and d L(G (1,1))(z 1 ), d L(G (1,1))(z 2 )>1. For integers a, b with a b 1, it is easily seen that L(G (a, b)) can be obtained from L(G (1, 1)) by attaching two pendent paths on a 1 and b 1 ertices, respectiely, to z 1 and z 2, i.e., L(G (a, b)) = L(G (1, 1)) z1,z 2 (a 1, b 1). If t 1, then by Corollary 4.1, EE(L(G (1, 1)) z1,z 2 (s 1, t 1)) < EE(L(G (1, 1)) z1,z 2 (s 2, t)), i.e., EE(L(G (s, t))) < EE(L(G (s 1, t + 1))). Sppose that t = 0. Let L(G (1, 0)) = L(G (1, 1)) z 2.ThenL(G (s, 0)) can be obtained from L(G (1, 0)) by attaching a pendent path on s 1erticestoz 1, i.e., L(G (s, 0)) = L(G (1, 0)) z1 (s 1, 0). Recall that L(G (s 1, 1)) = L(G (1, 1)) z1,z 2 (s 2, 0). Let N L(G (1,1))(z 1 ) \{z 2 }=N L(G (1,1))(z 2 ) \{z 1 }={w 1, w 2,...,w r }. It is easily seen that and ths L(G (1, 0)) z1 (s 1, 0) = L(G (1, 1)) z1,z 2 (s 2, 0) {z 1 w 1, z 1 w 2,...,z 1 w r }, EE(L(G (1, 0)) z1 (s 1, 0)) < EE(L(G (1, 1)) z1,z 2 (s 2, 0)), i.e., EE(L(G (s, 0))) < EE(L(G (s 1, 1))).

9 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) Minimm Laplacian Estrada index of trees with gien maximm degree Let D n, be the tree obtained by attaching 1 pendent ertices to one end ertex of the path P n +1, where 2 n 1. Theorem 6.1. Let G be an n-ertex tree with maximm degree,where2 n 1.ThenLEE(G) LEE(D n, ) with eqality if and only if G = D n,. Proof. The case = 2 is triial. Sppose in the following that 3 n 1. Let G be a tree with minimm Laplacian Estrada index among n-ertex trees with a ertex, say x, ofmaximmdegree. If there is another ertex in G different from x with degree at least three, then by Theorem 5.1,wemay get a tree of maximm degree with smaller Laplacian Estrada index, a contradiction. Ths, x is the niqe ertex in G with degree at least three, i.e., G is a tree obtained by attaching paths to a single ertex x.nowbytheorem5.1, wehaeg = D n,. Ilić and Zho [25] showed that the path is the niqe tree with minimm Laplacian Estrada index. By Theorem 5.1, LEE(D n, 1 )<LEE(D n, ) for 4 n 1. Together with Theorem 6.1, we hae Theorem 6.2. Let G be an n-ertex tree different from D n,3 and P n,wheren 5. ThenLEE(G) > LEE(D n,3 )>LEE(P n ). 7. Maximm Laplacian Estrada indices of trees with gien parameters First we gie some lemmas which will be sed in or proof. Lemma 7.1. Let G and G 1 be two trees shown in Fig. 3, where the path from to w in G is a pendent path at, and all neighbors of in Q of G are switched to be neighbors of in Q of G 1.Ifd G (, w) max{d G (, x) : x V(S)} and S is not a path with an end ertex, then LEE(G) <LEE(G 1 ). Proof. By Lemma 5.1, we need only to show that EE(L(G)) < EE(L(G 1 )). Let P be the path from to w in G. Sinced G (, w) max{d G (, x) : x V(S)}, thereisaertex z V(S) sch that d G (, w) d G (, z), i.e., d P (, w) d S (, z). SinceS is not a path with an end ertex, P is a proper sbgraph of S. Let 1 be the neighbor of in G lying on P ( 1 = w if and w are adjacent in G), and 1 be the neighbor of in G lying on the niqe path connecting and z ( 1 = z if and z are adjacent in G). Denote by w 1, w 2,...,w t the neighbors of in Q of G. For y 1 y 2 E(G), letx y1 y 2 be the ertex in L(G) corresponding to y 1 y 2 E(G). Let E x1 ={x 1 x w1, x 1 x w2,...,x 1 x wt }, E x1 ={x 1 x w1, x 1 x w2,...,x 1 x wt }. Fig. 3. The trees G and G 1 in Lemma 7.1.

10 2074 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) Fig. 4. The trees G 1 and G 2 in Lemma 7.2. Consider H = L(G) E x1. By similar proof of Lemma 4.2 (i) and (ii), we hae (H; x 1 ) (H; x 1 ), and (H; x wi, x 1 ) (H; x wi, x 1 ) for 1 i t. Note that L(G) = H + E x1.letg = H + E x1. It follows from Lemma 3.2 that EE(L(G)) < EE(G ). It is easily seen that G is a sbgraph of L(G 1 ), and ths EE(G ) EE(L(G 1 )), implying that EE(L(G)) < EE(L(G 1 )). Lemma 7.2 [25]. Let be a ertex of a tree Q with at least two ertices. For integer a 1, letg 1 be the tree obtained by attaching a star S a+1 atitscentertoofq,andg 2 be the tree obtained by attaching a + 1 pendent ertices to of Q, see Fig. 4. ThenLEE(G 1 )<LEE(G 2 ). Let D n,d bethetreeobtainedfromp d+1 = 0 1 d by attaching n d 1 pendent ertices to d/2, where 2 d n 1. Theorem 7.1. Let G be an n-ertex tree with diameter d, where 2 d n 1.ThenLEE(G) LEE(D n,d ) with eqality if and only if G = D n,d. Proof. The case d = n 1 is triial. Sppose that d < n 1. Let G be a tree with maximm Laplacian Estrada index among the n-ertex trees with diameter d, and P = 0 1 d be a diametrical path of G. By Lemma 7.2, eery ertex otside P is a pendent ertex. Let V 1 (G) be the set of ertices on P with degree at least three in G. Obiosly, V(G) 1sinced { < n 1. } Sppose that V 1 (G) 2. Assme that V 1 (G) 1, 2,..., d 1 =. Choose i V 1 (G) sch that d G ( i, 0 ) is as small as possible. Then d G ( i+1, d ) d G ( i, 0 ). Note that the component of G i i+1 containing i+1 is not a path with an end ertex i+1 since V 1 (G) 2. Now applying Lemma 7.1 to G by setting = i+1, = i and w = 0,wemaygetanothern-ertex tree with diameter d with larger Laplacian Estrada index, a contradiction. Ths, V 1 (G) = 1, i.e., G is a tree obtained by attaching n d 1 pendent ertices to s for 1 s d. It follows from Theorem that s = d 2, i.e., G = D n,d. Let n, p be positie integers. Let s = n 1 p, r = n 1 ps.lett n,p be the tree obtained by attaching p r paths on s ertices and r paths on s + 1 ertices to a single ertex, where 2 p n 1. Theorem 7.2. Let G be an n-ertex tree with p pendent ertices, where 2 p n 1. ThenLEE(G) LEE(T n,p ) with eqality if and only if G = T n,p. Proof. The cases p = 2, n 1 are triial. Sppose in the following that 3 p n 2. Let G be a tree with maximm Laplacian Estrada index among the n-ertex trees with p pendent ertices. Let V 1 (G) be the set of ertices in G with degree at least three. Let P be a pendent path with minimm length in G at a ertex V 1 (G), and w be the pendent ertex in P. Sppose that V 1 (G) 2. Choose a ertex y V 1 (G) sch that d G (, y) is as small as possible. Then the internal ertices (if exist) of the niqe path connecting and y in G are all of degree two. Denote by the neighbor of in G lying on the niqe path connecting and y ( = y if and y are adjacent in G). Let S be the component of G containing. Obiosly,S is not a path with an end ertex 2

11 Z. D, Z. Li / Linear Algebra and its Applications 435 (2011) since y V(S). By the choice of P, wehaed G (, w) max{d G (, x) : x V(S)}. Applying Lemma 7.1 to G, wemaygetanothern-ertex tree G 1 with p pendent ertices sch that LEE(G) <LEE(G 1 ), a contradiction. Ths V 1 (G) =1, i.e., G is a tree obtained by attaching p pendent paths to a single ertex. It follows from Theorem 5.1 that G = T n,p. Lemma 7.3. For 2 p n 2, LEE(T n,p )<LEE(T n,p+1 ). Proof. Let be the pendent ertex of a longest pendent path in T n,p.let be the neighbor of, and w be the neighbor of different from in T n,p.letg = T n,p + w.bytheorem5.1, LEE(T n,p )<LEE(G). Note that there are p + 1 pendent ertices in G, and ths by Theorem 7.2, LEE(G) LEE(T n,p+1 ).Then the reslt follows easily. A matching of a graph is an edge sbset in which no pair shares a common ertex. The matching nmber of G, denotedbym(g), is the maximm cardinality of a matching of G. For 1 r n/2, lett n,r be the tree obtained by attaching r 1pathsontwoerticestothe center of the star S n 2r+2. Corollary 7.1. Let G be a tree with n ertices and matching nmber m = m(g), where2 m n/2. Then LEE(G) LEE(T n,m ) with eqality if and only if G = T n,m. Proof. Let M be a maximm matching of G. Letp be the nmber of pendent ertices in G. Obiosly, there is at most one pendent end ertex for an edge of M. Thenp m + (n 2m) = n m. If p = n m, then by Theorem 7.2 (with s = 1 and r = m 1),wehaeT n,m = Tn,n m is the niqe tree with maximm Laplacian Estrada index. If p < n m, then by Theorem 7.2 and Lemma 7.3, LEE(G) LEE(T n,p )< < LEE(T n,n m ) = LEE(T n,m ). Then the reslt follows easily. An independent set of a graph is a ertex sbset in which no pair is adjacent. The independence nmber of a graph G, denotedbyα(g), is the maximm cardinality of an independent set of G. Itis well-known that for any bipartite graph G, α(g) + m(g) = V(G), see[2]. From Corollary 7.1, we hae Corollary 7.2. Let G be a tree with n ertices and independence nmber α = α(g),where n/2 α n 2. ThenLEE(G) LEE(T n,n α ) with eqality if and only if G = T n,n α. A dominating set of a graph is a ertex sbset whose closed neighborhood contains all ertices of the graph. The domination nmber of a graph G, denotedbyγ(g), is the minimm cardinality of a dominating set of G. AcoeringofagraphG is a ertex sbset K sch that eery edge of G has at least one end ertex in K. Let G be a tree. By König s theorem [2], the matching nmber of G is eqal to the minimm cardinality of a coering of G. It is easily seen that a coering of G is also a dominating set of G. It follows that m(g) γ(g).bytheorem5.1, LEE(T n,m )<LEE(T n,m 1 ) for 2 m n/2. Together with Corollary 7.1,wehae Corollary 7.3. Let G be a tree with n ertices and domination nmber γ = γ(g),where2 γ n/2. Then LEE(G) LEE(T n,γ ) with eqality if and only if G = T n,γ. Acknowledgements The athors are gratefl for the sefl comments and sggestions of the reiewers, which hae helped to improe the paper.

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