On the Estrada index of cacti

Abstract

Let G be a simple connected graph on n vertices and λ_1, λ_2, . . . , λ_n be the eigenvalues of the adjacency matrix of G. The Estrada index of G is defined as EE(G) = \sum_{i=1}^n e^{λi}. A cactus is a connected graph in which any two cycles have at most one common vertex. In this work, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cycles was determined. Also, the unique graph with maximal Estrada index in the class of all cacti with n vertices and k cut edges was determined.