The Minimal Estrada Index of Trees with Two Maximum Degree Vertices

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1 MATCH Commnications in Mathematical and in Compter Chemistry MATCH Commn. Math. Compt. Chem. 64 (2010) ISSN The Minimal Estrada Index of Trees with Two Maximm Degree Vertices Jing Li 1, Xeliang Li 1, Lsheng Wang 2 1 Center for Combinatorics and LPMC-TJKLC Nankai Uniersity, Tianjin , China lj02013@163.com; lxl@nankai.ed.cn 2 Department of Compter Science, City Uniersity of Hong Kong Tat Chee Aene, Kowloon, Hong Kong SAR, China cswangl@city.ed.hk (Receied December 22, 2009) Abstract Let G be a simple graph with n ertices and let λ 1,λ 2,...,λ n be the eigenales of its adjacency matrix. The Estrada index EE of G is the sm of the terms e λi. In 2009 Ilić et al. obtained the trees with minimal Estrada index among trees with a gien maximm ertex degree. In this paper, we gie the trees with minimal Estrada index among the trees of order n with exactly two ertices of maximm degree. 1 Introdction Let G be a simple graph with n ertices and m edges. A walk [1] in a simple graph G is a seqence W := 0 1 l of ertices, sch that i 1 i is an edge in G. If 0 = x and l = y, we say that W connects x to y and refer to W as an xy-walk. The ertices x and

2 -800- y are called the ends of the walk, x being its initial ertex and y its terminal ertex, while the ertices 1,..., l 1 are called its internal ertices. The integer l is the length of W. If and are two ertices of a walk W, where precedes on W, the sbseqence of W starting from and ending at is denoted by W and called the segment of W from to. The spectrm of G is the spectrm of its adjacency matrix [2], and consists of the (real) nmbers λ 1 λ 2 λ n. The Estrada index is defined as n EE(G) = e λi. Althogh inented in the year 2000 [6], the Estrada index has already fond a large nmber of applications, sch as in biochemistry [6, 7, 10] and in the theory of complex networks [8, 9]. Also nmeros lower and pper bonds for the Estrada index hae been commnicated [5, 11, 12, 15]. Recently, Deng in [3] showed that the path P n and the star S n hae the minimal and the maximal Estrada indices among n-ertex trees. Zhao and Jia in [14] determined also the trees with the second and the third maximal Estrada index. Then, Deng in [4] gae the first six trees with the maximal Estrada index. In 2009 Ilić et al. [13] obtained the trees with minimal Estrada index among trees of order n with a gien maximm ertex degree. In this paper, we gie the trees with minimal Estrada index among trees of order n with exactly two ertices of maximm degree. i=1 2 Preliminaries In or proofs, we will se a relation between EE and the spectral moments of a graph. For k 0, we denote by M k the k-th spectral moment of G, n M k (G) = λ k i. We know from [2] that M k is eqal to the nmber of closed walks of length k of the graph G, and the first few spectral moments of a graph with m edges and n ertices satisfy the relations: M 0 = n, M 1 =0, M 2 =2m, M 3 =6t, i=1

3 -801- where t is the nmber of triangles in G. From the Taylor expansion of e x, we hae the following important relation between the Estrada index and the spectral moments of G: EE(G) = k=0 M k k!. Ths, if for two graphs G and H we hae M k (G) M k (H) for all k 0, then EE(G) EE(H). Moreoer, if the strict ineqality M k (G) >M k (H) holds for at least one ale of k, then EE(G) >EE(H). 3 The minimal Estrada index of trees with two maximm degree ertices Let G i be a graph with n ertices and m edges, in which there are exactly two ertices of maximm degree Δ. Let W (G i ) denote the set of closed walks in G i, and W 2k (G i ) denote the set of closed walks of length 2k in G i. We say a closed walks is at ertex j, if it is a closed walk from j to j. Lemma 3.1 [3] Let be a non-isolated ertex of a simple graph H. If H 1 and H 2 are the graphs obtained from H by identifying an end ertex 1 and an internal ertex t of the n-ertex path P n, respectiely, with, see Figre 3.1, then M 2k (H 1 ) <M 2k (H 2 ) for n 3 and k 2. H 1 t t+1 n H 1 t t+1 n H 1 H 2 Figre 3.1 The transformation in Lemma 3.1. Lemma 3.2 For the two trees G 1 and G 2 in Figre 3.2, we hae EE(G 1 ) >EE(G 2 ), where P i are paths of length n i, n i 0, 1 i s, A and B are (connected) trees and G 1 G 2.

4 -802- A P 1 P 2 P 3 P s s G 1 B A s s+1 s + s i=1 ni B G 2 Figre 3.2 The graphs G 1 and G 2 in Lemma 3.2. Proof. Sppose P i is the first path with n i > 0. Let n i = m and i = i 0. Then G 1 can be redrawn as in Figre 3.3. First we show that if G 1 G 3, then EE(G 1 ) >EE(G 3 ). Consider the following correspondence: ξ : W 2k (G 3 ) W 2k (G 1 ), w W 2k (G 3 ). Denote by A 1, B 1 the graphs in Figre 3.3. Since W 2k (G i )=W 2k (A 1 ) W 2k (B 1 ) W i, where W i is the set of closed walks of length 2k of G i, each of which contains at least one edge in E(A 1 ) and at least one edge in E(B 1 ). So M 2k (G i )= W 2k (A 1 ) + W 2k (B 1 ) + W i = M 2k (A 1 )+M 2k (B 1 )+ W i. Obiosly, it is sfficient to show that W 1 > W 3. For any closed walk w W 3, it contains the segments w 1l of the walk in W (A 1 ), and the segments w 2j in W (B 1 ), 1 l, j t, t = max{t 1,t 2 }, where t 1 and t 2 are the nmbers of segments of w in W (A 1 ) and W (B 1 ), respectiely, and some of the segments may be empty. Then it can be written as w = w 11 w 21 w 12 w 22 w 1t w 2t. For the segments w 1l W (A 1 ), define ξ(w 1l )=w 1l, 1 l t. Now we define ξ(w 2j ), 1 j t. Let f : {i 0,i 1,...,i m } {i 0,i 1,...,i m }, f(i r )=i m r,for0 r m. Case 1. If w 2j does not pass the edge e i0,i+1, then define ξ(w 2j )=f(w 2j ). Case 2. If on the contrary, w 2j pass the edge e i0,i+1, then if w begins with a ertex in A 1, then w 2j is a closed walk at i m. It contains the first segment w 2j from the initial ertex i m to the first i 0, the second segment w 2j from the first i 0 to the last i 0, and the third segment w 2j from the last i 0 to the terminal ertex i m. Then, define ξ(w 2j )=f(w 2j) (w 2j) 1 w 2j, where (w 2j) 1 is the walk

5 -803- i m i 2 i 1 A 1 2 i 1 i(i 0) i +1 s G 1 B A 1 2 i 1 i m i 1 i 0 i +1 s B G 3 A 1 2 i 1 i(i m) i m i 1 i 0 i +1 s B A 1 B 1 Figre 3.3 The graphs in the proof of Lemma 3.2. from w 2j by reersing the order of all the ertices. It is the walk in G 1 that consists of the first segment from i 0 to the first i m, the second segment from i m to the next first i 0 and the third segment from i 0 to i 0. if w begins with a ertex except i m in B 1, and j 1,t, then define ξ(w 2j ) is the same as aboe. if w begins with a ertex except i m in B 1, j =1,t. We can see that w 11 is empty, w 21 contains the first segment w 21 from the initial ertex to the first i 0, the second segment w 21 from the first i 0 to the last i 0, and the third segment w 21 from the last i 0 to the terminal ertex i m. Then, define ξ(w 21 )=f(w 21) (w 21) 1 w 21. Andw 2t contains the first segment w 2t from i m to the first i 0, the second segment w 2t from the first i 0 to the last i 0, and the third segment w 2t from the last i 0 to the terminal ertex. Then, define ξ(w 2t )=w 2t (w 2t) 1 f(w 2t). We then define ξ(w) =ξ(w 11 ) ξ(w 21 ) ξ(w 12 ) ξ(w 22 ) ξ(w 1t ) ξ(w 2t ), for ξ(w) W 1.

6 -804- Now, for any closed walk w W 3, there is a niqe walk ξ(w) inw 1 corresponding to it. By the definition of ξ and the description aboe, we know that if there is a walk ξ(w) W 1, then ξ(w) can be diided into some pieces in only one way, and ξ on each piece is bijectie. We combine all the inerse images of the pieces according to the only order, so we can get the only w W 3. Therefore, if w 1,w 2 W 3,w 1 w 2, then ξ(w 1 ) ξ(w 2 ). Ths, ξ is injectie. Bt it is not srjectie, since there is no w W 2k (G 3 ) sch that ξ(w) =(i 1)i 0 (i +1)i 0 (i 1). Then clearly W 1 > W 3. Ths, M 2k (G 1 ) >M 2k (G 3 ), that is, EE(G 1 ) >EE(G 3 ). Then, we can repeat the aboe process and finally get that EE(G 1 ) >EE(G 2 ), as reqired. For any tree T of order n with exactly two ertices, of maximm degree Δ, by sing the transformation in Lemma 3.1 repeatedly, we can easily get that EE(T ) EE(G 1 ), where A denotes the nion of the Δ 1 disjoint paths all of which hae their end ertices adjacent to, and B denotes the nion of the Δ 1 disjoint paths all of which hae the end ertices adjacent to. Then from Lemma 3.2, EE(T ) EE(G 1 ) EE(G 2 ), and the eqality holds if and only if T = G 1 = G2. The following is a ery sefl lemma from [13]. Lemma 3.3 [13] Let w be a ertex of a nontriial connected graph G, and for nonnegatie integers p and q, let G(p, q) denote the graph obtained from G by attaching pendent paths P = w 1 2 p and Q = w 1 2 q of lengths p and q, respectiely, at w. Ifp q 1, then EE(G(p, q)) >EE(G(p +1,q 1)). Then by applying the transformation in Lemma 3.3 repeatedly, we get that EE(G 2 ) > EE(G 4 )ifg 2 G 4, where G 4 is the graph introdced in the following lemma. Lemma 3.4 Let G 4 and G 5 be two trees with n ertices, see Figre 3.4, we hae EE(G 4 ) > EE(G 5 ), where P i are paths of length n i, n i 1,i =1, 2, 3, and, are ertices with maximm degree Δ, and G 4 G 5. Proof. We first show that if G 4 G 6, then EE(G 4 ) >EE(G 6 ).

7 -805- P n3 1 P1 2 2 P2 n1 G 4 n i=1 n i 2 G n3 n 2 + n 3 1 P1 2 n1 G 6 Figre 3.4 The graphs G i, i =4, 5, 6 in Lemma 3.4. Since M 2k (G i )= W 2k (A) + W 2k (B) + W i = M 2k (A) +M 2k (B) + W i, where W i is the set of closed walks of length 2k of G i containing at least one edge in E(A) and at least one edge in E(B), i =4, 6. Similar to Lemma 3.2, we only need to show that W 4 > W 6. For conenience, we relabel B as B, m = n 2 + 2, see Figre 3.5. Now we show that for a closed walk w W 6, there is an injection ξ, sch that ξ(w) W 4. For any closed walk w W 6, it contains the segments w 1l of the walk in W (A), and the segments w 2j in W (B ), 1 l, j s, s = max{s 1,s 2 }, where s 1 and s 2 are the nmbers of segments of w in W (A) and W (B ), respectiely, and some of the segments may be empty. Then it can be written as w = w 11 w 21 w 12 w 22 w 1s w 2s. For the segments w 1l W (A), define ξ(w 1l )=w 1l, 1 l s. Now we define ξ(w 2j ), 1 j s. Case 1. w 2j only ses edges on the path P = m. Let f : { 1, 2,..., m } { 1, 2,..., m },f( i )= m+1 i, 1 i m. Then, define ξ(w 2j )=f(w 2j ). Case 2. w 2j also ses other edges of B.

8 -806- P P n 3 n 2 2 B n 1 A m B 3 2 Δ 2 Figre 3.5 The trees in the proof of Lemma 3.4. We first define the term stable segment S, it is a maximal consectie sbseqence of w 2j from i to 2,1 i Δ 2, all the edges of the sbseqence are of the form 2 k, 1 k Δ 2. Now we consider the remaining sbseqence w 2j of w 2j by deleting all the stable segments. Let f : { 2, 3..., m } { 1, 2,..., m 1 }, f ( i ) = i 1, 2 i m. f : { 2,..., m } { 2,..., m }, f ( i )= m+2 i, 2 i m. w 1 is the walk from w by reersing the order of all the ertices. Sbcase 2.1. If w begins with a ertex in A, orw begins with a ertex in B and j 1,s, then w 2j is a closed walk at m. We only need to show that ξ(w 2j ) is a closed walk at 1. Actally, it is easy to see that w 2j is also a closed walk at m. If w 2j passes the ertex 1,soisw 2j. Thenw 2j consists of for segments: the first segment ŵ 1 from the initial ertex m to the first 2, the second segment ŵ 2 from the first 2 to the 2 that is jst before the first 1, the third segment ŵ 3 = 2 1, where the 1 is the first 1 in w 2j, and the forth segment ŵ 4 from the first 1 to the terminal ertex of w 2j. Actally, some of the segments may be empty. Let Si t be the stable segment after the i-th 2 of ŵ t,1 t 4. Let ξ(ŵ 1)=f (ŵ 1), it is a walk from 2 to m, no internal ertices is 1 or m. And ξ(ŵ 1 ) is the walk from ξ(ŵ 1) by inserting S1 1 (if it exists) after the first 2 in it. Let ξ(ŵ 2)=f (ŵ 2), it is a walk from 1 to 1, and no internal ertices is m. Since in ŵ 2, there mst be a 3 before each 2 except the first one, and f ( 3 )= 2,f ( 2 )= 1. So we can define ξ(ŵ 2 ) to be the walk from ξ(ŵ 2) by inserting Si 2 after the 2 that is jst before the i-th 1, i 2. Let ξ(ŵ 3 )=ξ(ŵ 3)=(ŵ 3) 1 = 1 2. Finally, let ξ(ŵ 4)=(ŵ 4) 1,it

9 -807- is a walk from m to 1, and ξ(ŵ 4 ) is the walk from ξ(ŵ 4) by inserting all Si 4 to the original place in w 2j. Ths, we define ξ(w 2j )=ξ(ŵ 2 ) ξ(ŵ 3 ) ξ(ŵ 1 ) ξ(ŵ 4 ), it is a closed walk at 1. On the other hand, if ξ(w 2j ) is gien, we can get the for parts niqely according to the featres we described aboe. If w 2j does not pass the ertex 1, it mst pass the ertex 2, and so is w 2j. Then w 2j consists of three segments: the first segment ŵ 1 from the initial ertex m to the first 2, the second segment ŵ 2 from the first 2 to the last 2, and the third segment ŵ 3 from the last 2 to the terminal ertex of w 2j. Let Si t be the stable segment after the i-th 2 of ŵ t,1 t 3. Let ξ(ŵ 1)=f(ŵ 1), it is a walk from 1 to m 1, no internal ertices is m 1. And ξ(ŵ 1 ) is the walk from ξ(ŵ 1) by inserting S1 1 (if it exists) after the first 2 in it. Let ξ(ŵ 2)=f (ŵ 2), it is a walk from 1 to 1. Since in ŵ 2, there mst be a 3 before each 2 except the first one, and f ( 3 )= 2,f ( 2 )= 1. So we can define ξ(ŵ 2 ) to be the walk from ξ(ŵ 2) by inserting Si 2 after the 2 that is jst before the i-th 1, i 2. Finally, let ξ(ŵ 3 )=ξ(ŵ 3)=(f (ŵ 3)) 1, it is a walk from m 1 to 1, no internal ertices is 1. Ths, we define ξ(w 2j )=ξ(ŵ 1 ) ξ(ŵ 3 ) ξ(ŵ 2 ), it is a closed walk at 1. On the other hand, if ξ(w 2j ) is gien, we can get the three parts niqely according to the featres we described aboe. Sbcase 2.2. If w begins with a ertex in B, and j =1,s, the for the two cases that w 2j,j =1,s passes the ertex 1 or does not pass 1, both can be defined similarly as aboe. Ths, If w begins and ends with ertex t, 2 t m 1, ξ(w 21 ) can be defined niqely to be a walk from m+1 t to 1, ξ(w 2s ) can be defined niqely to be a walk from 1 to m+1 t.ifwbegins and ends with ertex t, 1 t Δ 2, ξ(w 21 ) can be defined niqely to be a walk from m 1 to 1, ξ(w 2s ) can be defined niqely to be a walk from 1 to m 1. Then we define ξ(w) =ξ(w 11 ) ξ(w 21 ) ξ(w 12 ) ξ(w 22 ) ξ(w 1s ) ξ(w 2s ), for ξ(w) W 4. Now, for any closed walk w W 6, there is a niqe walk ξ(w) inw 4 corresponding to it. By the definition of ξ and the description aboe, we know if there is a walk ξ(w) W 4, ξ(w) can be diided into some pieces in only one way, and ξ on each piece is bijectie. We combine all the inerse images of the pieces according to the only order, so we can

10 -808- get the only w W 6. Therefore, if w 1,w 2 W 6,w 1 w 2, then ξ(w 1 ) ξ(w 2 ). Ths, ξ is injectie. Bt it is not srjectie, since there is no w W 6, sch that there is a segment ξ(w 2j ) ξ(w) with ξ(w 2j )= , 1 j s. Ths, W 4 > W 6, and conseqently, M 2k (G 4 ) > M 2k (G 6 ), that is, EE(G 4 ) > EE(G 6 ). Analogosly, we can get that EE(G 6 ) > EE(G 5 )ifg 5 G 6. Ths EE(G 4 ) > EE(G 5 ), as reqired. The aboe lemma is tre for the case n 3 1, which means that and are not adjacent. Actally the lemma is also tre when and are adjacent. We can proe it similarly. From Lemmas 3.1 throgh 3.4, we finally get the reslt below. Theorem 3.5 For all trees T of order n with exactly two ertices of maximm degree, the graph G 5 has the minimal Estrada index. With one more restriction that the two maximm degree ertices of the trees mst be adjacent, we gie the following conjectre. Conjectre 3.6 For all trees T of order n with two adjacent ertices of maximm degree, the graph G 7 has the minimal Estrada index, see Figre 3.6, where, are ertices with maximm degree. Theorem 3.5 can be generalized to trees with one maximm and one second maximm degree ertex as follows. Theorem 3.7 For all trees T of order n with exactly one maximm and one second maximm degree ertex, the graph G 8 has the minimal Estrada index, see Figre 3.7, where, are ertices with the maximm and second maximm degree, respectiely. Acknowledgement: The athors Jing Li and Xeliang Li are spported by NSFC No , PCSIRT and the 973 program. The athor Lsheng Wang is flly spported by a grant from the Research Grants Concil of the Hong Kong SAR [Project No. CityU ]. The athors are ery gratefl to the referees for helpfl comments and sggestions.

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