On The Least Signless Laplacian Eigenvalue of Non-Bipartite Unicyclic Graphs with Both Given Order and Diameter

Abstract

Let $A$ be the $(0,1)$-adjacency matrix of a simple graph $G$, and $D$ be the diagonal matrix $diag(d_1,d_2,\dots ,d_n)$, where $d_i$ is the degree of the vertex $v_i$. The matrix $Q(G)=D+A$ is called the signless Laplacian of $G$. In this paper, we characterize the extremal graph for which the least signless Laplacian eigenvalue attains its minimum among all non-bipartite unicyclic graphs with given order and diameter.