Note on unicyclic graphs with given number of pendent vertices and minimal energy

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let G(n,p) denote the set of unicyclic graphs with n vertices and p pendent vertices. In [H. Hua, M. Wang, Unicyclic graphs with given number of pendent vertices and minimal energy, Linear Algebra Appl. 426 (2007) 478–489], Hua and Wang discussed the graphs that have minimal energy in G(n,p), and determined the minimal-energy graphs among almost all different cases of n and p. In their work, certain case of the values was left as an open problem in which the minimal-energy species have to be chosen in two candidate graphs, but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials. This paper aims at solving the problem completely, by using the well-known Coulson integral formula.