Some Tur\'{a}n-type results for the signless Laplacian spectral radius

TL;DR

This paper extends classical extremal graph results by exploring the signless Laplacian spectral radius, providing new supersaturation, blowup, and stability results that generalize known spectral theorems.

Contribution

It introduces variants of Nikiforov's spectral results using the signless Laplacian, broadening the understanding of spectral extremal graph theory.

Findings

Extended supersaturation results for signless Laplacian spectral radius

Established blowup of cliques under signless Laplacian conditions

Provided stability results related to spectral thresholds

Abstract

Half a century ago, Bollob\'{a}s and Erd\H{o}s [Bull. London Math. Soc. 5 (1973)] proved that every $n$-vertex graph $G$ with $e(G)\ge (1- \frac{1}{k} + \varepsilon )\frac{n^2}{2}$ edges contains a blowup $K_{k+1}[t]$ with $t=\Omega_{k,\varepsilon}(\log n)$. A well-known theorem of Nikiforov [Combin. Probab. Comput. 18 (3) (2009)] asserts that if $G$ is an $n$-vertex graph with adjacency spectral radius $\lambda (G)\ge (1- \frac{1}{k} + \varepsilon)n$, then $G$ contains a blowup $K_{k+1}[t]$ with $t=\Omega_{k,\varepsilon}(\log n)$. This gives a spectral version of the Bollob\'{a}s--Erd\H{o}s theorem. In this paper, we systematically explore variants of Nikiforov's result in terms of the signless Laplacian spectral radius, extending the supersaturation, blowup of cliques and the stability results.