On bicyclic graphs with maximal energy

AbstractLet λ1,λ2,…,λn be the eigenvalues of a graph G of order n. The energy of G is defined as E(G)=|λ1|+|λ2|+⋯+|λn|. Let Pn6,6 be the graph obtained from two copies of C6 joined by a path Pn-10, Bn be the class of all bipartite bicyclic graphs that are not the graph obtained from two cycles Ca and Cb (a,b⩾10 and a≡b≡2 (mod 4)) joined by an edge. In this paper, we show that Pn6,6 is the graph with maximal energy in Bn, which gives a partial solution to Gutman’s conjecture in Gutman and Vidović (2001) [I. Gutman, D. Vidović, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41 (2001) 1002–1005]