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

Shu-Guang Guo1, Meiling Xu1,2, Guanglong Yu1
1Department of Mathematics, Yancheng Teachers University, Yancheng, 224002, Jiangsu, P.R. China
2Department of Mathematics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, P.R. China

Abstract

Let \(A\) be the \((0,1)\)-adjacency matrix of a simple graph \(G\), and \(D\) be the diagonal matrix \(diag(d_1, d_2, \ldots, d_n)\), 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.