On the second largest eigenvalue of the signless Laplacian

Abstract

Let $G$ be a graph of order $n,$ and let $q_1(G)\ge ...\ge q_n(G)$ be the eigenvalues of the $Q$-matrix of $G$, also known as the signless Laplacian of $G.$ In this paper we give a necessary and sufficient condition for the equality $q_k(G)=n-2,$ where $1<k\le n.$ In particular, this result solves an open problem raised by Wang, Belardo, Huang and Borovicanin. We also show that [ q_{2}(G) \geqδ(G)] and determine that equality holds if and only if $G$ is one of the following graphs: a star, a complete regular multipartite graph, the graph $K_{1,3,3},$ or a complete multipartite graph of the type $K_{1,...,1,2,...,2}$.

arXiv commentThis version fills a gap in one proof, noticed by Rundan Xing

Cited by in corpus (3)

• On the $A_{\alpha }$-spectra of graphs
• Signless Laplacian eigenvalue problems of Nordhaus-Gaddum type
• Signless Laplacian spectral characterization of some graphs