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

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, Cvetković, Rowlinson, and Simić.