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 ) = ∑ i = 1 n e λ i , where λ 1 , λ 2 , … , λ n are the eigenvalues of the adjacency matrix of G , and the Kirchhoff index of a graph G is defined as Kf ( G ) = ∑ i < j r i j , where r i j is the resistance distance between vertices v i and v j 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 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 ) (resp. Kf ( G ‾ ) ) 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.