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The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum ofthe absolute values of all eigenvalues of its adjacency matrix. Let $C_n$denote the cycle of order $n$ and $P^{6,6}_n$ the graph obtained from joiningtwo cycles $C_6$ by a path $P_{n-12}$ with its two leaves. Let $\mathscr{B}_n$denote the class of all bipartite bicyclic graphs but not the graph $R_{a,b}$,which is obtained from joining two cycles $C_a$ and $C_b$ ($a, b\geq 10$ and $a\equiv b\equiv 2\, (\,\textmd{mod}\, 4)$) by an edge. In [I. Gutman, D.Vidovi\'{c}, Quest for molecular graphs with maximal energy: a computerexperiment, {\it J. Chem. Inf. Sci.} {\bf41}(2001), 1002--1005], Gutman andVidovi\'{c} conjectured that the bicyclic graph with maximal energy is$P^{6,6}_n$, for $n=14$ and $n\geq 16$. In [X. Li, J. Zhang, On bicyclic graphswith maximal energy, {\it Linear Algebra Appl.} {\bf427}(2007), 87--98], Li andZhang showed that the conjecture is true for graphs in the class$\mathscr{B}_n$. However, they could not determine which of the two graphs$R_{a,b}$ and $P^{6,6}_n$ has the maximal value of energy. In [B. Furtula, S.Radenkovi\'{c}, I. Gutman, Bicyclic molecular graphs with the greatest energy,{\it J. Serb. Chem. Soc.} {\bf73(4)}(2008), 431--433], numerical computationsup to $a+b=50$ were reported, supporting the conjecture. So, it is stillnecessary to have a mathematical proof to this conjecture. This paper is toshow that the energy of $P^{6,6}_n$ is larger than that of $R_{a,b}$, whichproves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclicgraphs, the conjecture is still open.

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