Complete solution to a problem on the maximal energy of unicyclic bipartite graphs

The energy of a simple graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by Cn the cycle, and Pn6 the unicyclic graph obtained by connecting a vertex of C6 with a leaf of Pn-6. Caporossi et al. conjectured that the unicyclic graph with maximal energy is Pn6 for n=8,12,14 and n≥16. In Hou et al. (2002) [Y. Hou, I. Gutman, C. Woo, Unicyclic graphs with maximal energy, Linear Algebra Appl. 356 (2002) 27-36], the authors proved that E(Pn6) is maximal within the class of the unicyclic bipartite n-vertex graphs differing from Cn. And they also claimed that the energies of Cn and Pn6 is quasi-order incomparable and left this as an open problem. In this paper, by utilizing the Coulson integral formula and some knowledge of real analysis, especially by employing certain combinatorial techniques, we show that the energy of Pn6 is greater than that of Cn for n=8,12,14 and n≥16, which completely solves this open problem and partially solves the above conjecture.