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

الأصل · EN

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. conjecture that the unicyclic graph with maximal energy is Pₙ⁶ for n=8,12,14 and n≥ 16. In``Y. Hou, I. Gutman and 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 energy 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.