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

Abstract

Let U r n 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 U r n. Firstly, among a subset of U r n in which each graph has a fixed odd girth ≥ 3, where n ≥ + 1 and r ≥ 1, we characterize a unique extremal graph which has the minimum SLCs and the minimum IE. Secondly, if G ∈ U r n and G has odd girth ≥ 5, where n ≥ 7 and r ≥ 1, then we prove that a unique extremal graph L n ∈ U r n with girth 4 satisfies that both the SLCs and the IE of G are more than the counterparts of L n .

The signless Laplacian coefficients and the incidence energy of graphs with a given bipartition

“…In the connected graphs of n vertices and m edges without even cycles, Wang et al [23] obtained the minimal element which has the minimum SLCs and the minimum IE. Among the unicyclic graphs with n vertices and r pendent vertices, where n ≥ 4 and r ≥ 1, Wang and Zhong [22] characterized a unique extremal graph which has the minimum SLCs and the minimum IE. For further information on the signless Laplacian matrix, one can refer to three surveys [2][3][4].…”