Unicyclic graphs with given number of pendent vertices and minimal energy

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 G(n,l,p) denote the set of all unicyclic graphs on n vertices with girth and pendent vertices being and , respectively. More recently, one of the present authors H. Hua, On minimal energy of unicyclic graphs with prescribed girth and pendent vertices, Match 57 (2007) 351–361] determined the minimal-energy graph in G(n,l,p). In this work we almost completely solve this problem, cf. Theorem 15. We characterize the graphs having minimal energy among all elements of G(n,p), the set of unicyclic graphs with n vertices and p pendent vertices. Exceptionally, for some values of n and p (see Theorem 15) we reduce the problem to finding the minimal-energy species to only two graphs.