The signless Laplacian coefficients and the incidence energy of unicyclic graphs with given pendent vertices

Abstract

Let Urn be the set of unicyclic graphs with n vertices and r pendent vertices (namely, r leaves), where n ≥ 4 and r ≥ 1. We consider the signless Laplacian coefficients (SLCs) and the incidence energy (IE) in Urn. Firstly, among a subset of Urn in which each graph has a fixed odd girth g ≥ 3, where n ≥ g + 1 and r ≥ 1, we characterize a unique extremal graph which has the minimum SLCs and the minimum IE. Secondly, if G  Urn and G has odd girth g ≥ 5, where n ≥ 7 and r ≥ 1, then we prove that a unique extremal graph Ln  Urn with girth 4 satisfies that both the SLCs and the IE of G are more than the counterparts of Ln.