Note on Bipartite Unicyclic Graphs of Given Bipartition with Minimal Energy

The energy of a graph G is defined as the sum of the absolute values of all the eigenvalues of the graph. Let UB(p,q) denote the set of all bipartite unicyclic graphs of a given (p, q)-bipartition, where q >= p >= 2. B(p, q) denotes the graph formed by attaching p - 2 and q - 2 vertices to two adjacent vertices of a quadrangle C(4), respectively, and H(3, q) denotes the graph formed by attaching q - 2 vertices to the pendent vertex of B(2, 3). In the paper "F. Li and B. Zhou, Minimal energy of bipartite unicyclic graphs of a given bipartition, MATCH Commun. Math. Comput. Chem. 54(2005), 379-388", the authors proved that either B(3, q) or H(3, q) is the graph with minimal energy in UB(3, q)(q >= 3). At the end of the paper they conjectured that H(3, q) achieves the minimal energy in UB(3,q) and checked that this is true for q = 3, 4. However, they could not find a proper way to prove it generally. This short note is to give a confirmative proof to the conjecture.