Some Turán-type results for the signless Laplacian spectral radius

Half a century ago, Bollobás and Erdős [Bull. London Math. Soc. 5 (1973)] proved that every $n$-vertex graph $G$ with $e(G)\ge (1-\frac{1}{k}+\epsilon )\frac{n^2}{2}$ edges contains a blowup $K_{k+1}[t]$ with $t={\mathrm{\Omega }}_{k,\epsilon }(\log n)$. 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 $\lambda (G)\ge (1-\frac{1}{k}+\epsilon )n$, then $G$ contains a blowup $K_{k+1}[t]$ with $t={\mathrm{\Omega }}_{k,\epsilon }(\log n)$. This gives a spectral version of the Bollobás–Erdős 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.