On extremal bipartite bicyclic graphs

Let B-n(+) be the set of all connected bipartite bicyclic graphs with n vertices. The Estrada index of a graph G is defined as EE(G) = Sigma(n)(i=1) e(lambda i), where lambda(1), lambda(2,) ... , lambda(n) are the eigemralues of the adjacency matrix of G, and the Kirchhoff index of a graph G is defined as Kf (G) = Sigma(i<j) rij, where rij is the resistance distance between vertices vi and vj in G. The complement of G. is denoted by <(G)over bar>. In this paper, sharp upper bound on EE(G) (resp. Kf ((G) over bar)) of graph G in B-n(+) is established. The corresponding extremal graphs are determined, respectively. Furthermore, by means of some newly created inequalities, the graph G in B-n(+) with the second maximal EE(G) (rasp. Kf((G) over bar)) is identified as well. It is interesting to see that the first two bicyclic graphs in B-n(+) according to these two orderings are mainly coincident. (C) 2015 Elsevier Inc. All rights reserved.