The least eigenvalue of the signless Laplacian of non-bipartite unicyclic graphs with k pendant vertices

Abstract

Let U(n,k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n � 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n,k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipartite unicyclic graphs on n vertices. As an application of the above results, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in Un is characterized, which has recently been proved by Cardoso, Cvetkovic, Rowlinson, and Simic. 1. Introduction. All graphs considered are simple, undirected, and connected. The vertex set and edge set of the graph G are denoted by V (G) and E(G), respec- tively. The distance between vertices u and v of a graph G is denoted by dG(u,v). The degree of a vertex v, written by dG(v) or d(v), is the number of edges incident with v. A pendant vertex is a vertex of degree 1. The set of the neighbors of a vertex v is denoted by NG(v) or N(v). The girth g(G) of a graph G is the length of the shortest cycle in G, with the girth of an acyclic graph being infinite. Denote by Cn