Estrada Index. Bo Zhou. Augest 5, Department of Mathematics, South China Normal University

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1 Outline 1. Introduction 2. Results for 3. References Bo Zhou Department of Mathematics, South China Normal University Augest 5, 2010

2 Outline 1. Introduction 2. Results for 3. References Outline 1. Introduction 2. Results for 3. References

3 Outline 1. Introduction 2. Results for 3. References Let G be a simple graph with vertex set V (G). The eigenvalues of G are the eigenvalues of its adjacency matrix A(G), denoted by λ 1,λ 2,...,λ n, where n = V (G).

4 Outline 1. Introduction 2. Results for 3. References The Estrada index of a graph G is defined as n EE(G) = e λ i. i=1

5 Outline 1. Introduction 2. Results for 3. References The Estrada index has been successfully employed to quantify the degree of folding of long-chain molecules, especially proteins, and to measure the centrality of complex (reaction, metabolic, communication, social, etc.) networks. There is also a connection between the Estrada index and the extended atomic branching of molecules.

6 Outline 1. Introduction 2. Results for 3. References Many properties of the Estrada index have been established. Various bounds for the Estrada index can be found in the literature. For example, the star S n is the unique n-vertex tree with maximum Estrada index, and the path P n is the unique n-vertex tree with minimum Estrada index. J.A. de la Peña, I. Gutman, J. Rada, Estimating the Estrada index, Linear Algebra Appl. 427 (2007) H. Deng, A proof of a conjecture on the Estrada index, MATCH Commun. Math. Comput. Chem. 62 (2009)

7 Outline 1. Introduction 2. Results for 3. References Ilić and Stevanović determined the unique tree with minimum Estrada index among the set of trees with given maximum degree. A. Ilić, D. Stevanović, The Estrada index of chemical trees, J. Math. Chem. 47 (2010) Li et al. determined the unique tree with minimum Estrada index among the set of trees with exactly two vertices of maximum degree. J. Li, X. Li, L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem. 64 (2010)

8 Outline 1. Introduction 2. Results for 3. References Theorem Let G be an (n,m)-graph with nullity n 0 < n. Then ( ) 2m EE(G) n 0 + (n n 0 )cosh n n 0 with equality if and only if n n 0 is even, G consists of copies of complete bipartite graphs K ri,t i, i = 1,2,..., n n 0 2, such that all r i t i are equal, and n n n 0 2 i=1 (r i + t i ) isolated vertices.

9 Outline 1. Introduction 2. Results for 3. References Theorem Let G be an (n,m)-graph. Then for any integer k 0 2, EE(G) n 1 k 0 M k (G) ( 2m ) k 2m + k! k=2 + e 2m (1) with equality if and only if G = K n. Setting k 0 = 2, 3, we have EE(G) n 1 2m + 2m e, ( EE(G) n m ) 2m + t + e 2m. 3

10 Outline 1. Introduction 2. Results for 3. References Let T(n,p) be the set of trees with n vertices and p pendant vertices, where 2 p n 1. Let n,p be positive integers. Let s = n 1 p, r = n 1 ps. Let T n,p be the tree obtained by attaching p r paths on s vertices and r paths on s + 1 vertices to a single vertex, where 2 p n 1. Theorem Let G T(n,p), where 2 p n 1. Then EE(G) EE(T n,p ) with equality if and only if G = T n,p.

11 Outline 1. Introduction 2. Results for 3. References For 2 r n/2, let T n,r be the tree obtained by attaching r 1 paths on two vertices to the center of the star S n 2r+2. Corollary Let G be a tree with n vertices and matching number m, where 2 m n/2. Then EE(G) EE(T n,m ) with equality if and only if G = T n,m. Corollary Let G be a tree with n vertices and independence number α, where n/2 α n 2. Then EE(G) EE(T n,n α ) with equality if and only if G = T n,n α. Corollary Let G be a tree with n vertices and domination number γ, where 2 γ n/2. Then EE(G) EE(T n,γ ) with equality if and only if G = T n,γ.

12 Outline 1. Introduction 2. Results for 3. References Let D n, be the tree obtained by adding an edge between the centers of two vertex-disjoint stars S, and attaching a path on n 2 vertices to a pendant vertex, where n Theorem Let G be an n-vertex tree with two adjacent vertices of maximum degree, where n Then EE(G) EE(D n, ) with equality if and only if G = D n,. This was conjectured in [J. Li, X. Li, L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem. 64 (2010) ].

13 Outline 1. Introduction 2. Results for 3. References Theorem Let G be a connected graph with n vertices and k cut edges, where 0 k n 3. Then EE(G) EE(G n,k ) with equality if and only if G = G n,k, where G n,k is the graph obtained from the complete graph on n k vertices by attaching k pendant edges to a vertex.

14 Outline 1. Introduction 2. Results for 3. References B. Zhou, On Estrada index, MATCH Commun. Math. Comput. Chem. 60 (2008) Z. Du, B. Zhou, The Estrada index of trees, Linear Algebra Appl. 435 (2011) Z. Du, B. Zhou, On the Estrada index of graphs with given number of cut edges, Electron. J. Linear Algebra 22 (2011)

15 Outline 1. Introduction 2. Results for 3. References Thank you!

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