Solution to a conjecture on the maximal energy of bipartite bicyclic graphs

Abstract: The energy of a simple graph , denoted by , is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let denote the cycle of order and the graph obtained from joining two cycles by a path with its two leaves. Let denote the class of all bipartite bicyclic graphs but not the graph , which is obtained from joining two cycles and ( and ) by an edge. In [I. Gutman, D. Vidović, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41(2001) 1002–1005], Gutman and Vidović conjectured that the bicyclic graph with maximal energy is , for and . In [X. Li, J. Zhang, On bicyclic graphs with maximal energy, Linear Algebra Appl. 427(2007) 87–98], Li and Zhang showed that the conjecture is true for graphs in the class . However, they could not determine which of the two graphs and has the maximal value of energy. In [B. Furtula, S. Radenković, I. Gutman, Bicyclic molecular graphs with the greatest energy, J. Serb. Chem. Soc. 73(4)(2008) 431–433], numerical computations up to were reported, supporting the conjecture. So, it is still necessary to have a mathematical proof to this conjecture. This paper is to show that the energy of is larger than that of , which proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, the conjecture is still open. [Copyright &y& Elsevier]