On the second largest eigenvalue of the signless Laplacian

TL;DR

This paper characterizes when the eigenvalues of the signless Laplacian of a graph equal specific values, solving an open problem and identifying extremal graphs for the second eigenvalue.

Contribution

It provides a necessary and sufficient condition for when the eigenvalues of the signless Laplacian equal n-2, addressing an open problem in spectral graph theory.

Findings

Characterization of when q_k(G) = n-2 for the signless Laplacian.

Proof that q_2(G) ≥ δ(G) with equality for specific graph classes.

Identification of extremal graphs for the second eigenvalue.

Abstract

Let $G$ be a graph of order $n,$ and let $q_{1}(G) \geq ...\geq 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 $1<k\leq n.$ In particular, this result solves an open problem raised by Wang, Belardo, Huang and Borovicanin. We also show that [ q_{2}(G) \geq\delta(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 $K_{1,...,1,2,...,2}$.