The clique number and the smallest Q-eigenvalue of graphs

TL;DR

This paper investigates the maximum smallest eigenvalue of the signless Laplacian in graphs without large cliques, extending classical bounds and establishing asymptotic results using graph blowups.

Contribution

It provides new bounds and asymptotic results for the smallest signless Laplacian eigenvalue in graphs avoiding large cliques, generalizing previous work.

Findings

Derived bounds on $q_{min}$ for graphs without $K_{r+1}$

Connected $q_{min}$ to bipartite graph properties

Established asymptotic maximum $q_{min}$ using blowup techniques

Abstract

Let $q_{\min}(G)$ stand for the smallest eigenvalue of the signless Laplacian of a graph $G$ of order $n.$ This paper gives some results on the following extremal problem: How large can $q_\min\left( G\right) $ be if $G$ is a graph of order $n,$ with no complete subgraph of order $r+1?$ It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds on $q_{\min}$ are obtained, thus extending previous work of Brandt for regular graphs. In addition, using graph blowups, a general asymptotic result about the maximum $q_{\min}$ is established. As a supporting tool, the spectra of the Laplacian and the signless Laplacian of blowups of graphs are calculated.