On ABC Estrada index of graphs

Abstract

Let G be a graph with vertex set VG={v1,v2,…,vn} and edge set EG, and let di be the degree of the vertex vi. The ABC matrix of G has the value (di+dj−2)/(didj) if vivj∈EG, and 0 otherwise, as its (i,j)-entry. Let γ1,γ2,…,γn be the eigenvalues of the ABC matrix of G in a non-increasing order. Then the ABC Estrada index of G is defined as EEABC(G)=∑i=1neγi and the ABC energy of G is defined as EABC(G)=∑i=1n|γi|. In this paper, some explicit bounds for the ABC Estrada index of graphs concerning the number of vertices, the number of edges, the maximum degree and the minimum degree, are established. Moreover, some bounds for the ABC Estrada index involving the ABC energy of graphs are also presented. All the corresponding extremal graphs are characterized respectively.