Some Turán-type results for the signless Laplacian spectral radius

Abstract

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 $t={\Omega }_{k,\epsilon }(\log n)$. 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 $t={\Omega }_{k,\epsilon }(\log n)$. 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.

31 pages. Any suggestions are welcome

References in corpus (8)

• Signless Laplacian spectral radius of graphs without short cycles or long cycles
• Counting substructures and eigenvalues I: triangles
• Signless Laplacian spectral radius and Hamiltonicity of graphs with large minimum degree
• A spectral Erdős-Faudree-Rousseau theorem
• Spectral supersaturation: Triangles and bowties
• A spectral Erdős-Rademacher theorem
• Linear spectral Turan problems for expansions of graphs with given chromatic number
• Spectral bipartite Turan problems on linear hypergraphs