On unicyclic graphs with given number of pendent vertices and minimal energy

Abstract

Let G(n,p) denote the set of unicyclic graphs with n vertices and p≥1 pendent vertices. Hua and Wang (2007) [4] claim to have determined the graph with minimal energy in G(n,p) except for the case when n=p+5 which was left as an open problem. This case was later solved by Huo et al. (2010) [5]. In this paper, we provide a counter example to show that the main result of Hua and Wang (2007) is not true. We also give the correct form of the characterization of unicyclic graphs with given number of pendent vertices and minimal energy. We make use of the well known Coulson's integral formula for the energy of graphs. Our proof is more general and also includes the case n=p+5 considered by Huo, Ji and Li (2010).