On extremal bipartite bicyclic graphs

Abstract

Let Bn+ be the set of all connected bipartite bicyclic graphs with n vertices. The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G, and the Kirchhoff index of a graph G is defined as Kf(G)=∑i<jrij, where rij is the resistance distance between vertices vi and vj in G. The complement of G is denoted by G‾. In this paper, sharp upper bound on EE(G) (resp. Kf(G‾)) of graph G in Bn+ is established. The corresponding extremal graphs are determined, respectively. Furthermore, by means of some newly created inequalities, the graph G in Bn+ with the second maximal EE(G) (resp. Kf(G‾)) is identified as well. It is interesting to see that the first two bicyclic graphs in Bn+ according to these two orderings are mainly coincident.