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

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.