Unicyclic graphs with given number of pendent vertices and minimal energy

Abstract

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 l ( ⩾ 3 ) and p ( ⩾ 1 ) , 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.