On minimal energy of bipartite unicyclic graphs of a given bipartition

The energy of a graph G, denoted by E(G), is defined to be the sum of absolute values of all eigenvalues of the adjacency matrix of G. Let B(p, q) denote the set of bipartite unicyclic graphs with a (p, q)-bipartition, where q >= p >= 2. Recently, Li and Zhou [MATCH Commun. Math. Comput. Chem. 54 (2005) 379-388.] conjectured that for q >= 3, E (B (3, q)) > E (H (3, q)), where B(3, q) and H(3, q) are respectively graphs as shown in Fig.1. In this note, we show that this conjecture is true for 3 <= q <= 217. As a byproduct, we determined the graph with minimal energy among all graphs in B(3, q).