The celebrated Erd\H{ o}s--Stone--Simonovits theorem states that $\mathrm{ ex}(n,F)= \big(1-\frac{ 1}{ \chi(F)-1}+o(1) \big)\frac{ n^{ 2}}{ 2}$, where $\chi(F)$ is the chromatic number of $F$. In 2009, Nikiforov proved a spectral extension of the Erd\H{ o}s--Stone--Simonovits theorem in terms of the adjacency spectral radius. In this paper, we shall establish a unified extension in terms of the signless Laplacian spectral radius. Let $q(G)$ be the signless Laplacian spectral radius of $G$ and we denote $\mathrm{ ex}_{ q}(n,F) =\max \{ q(G):|G|=n ~\mbox{ and}~F\nsubseteq G\}$. It is known that the Erd\H{ o}s--Stone--Simonovits type result for the signless Laplacian spectral radius does not hold for even cycles. We prove that if $F$ is a graph with $\chi(F)\geq 3$, then $\mathrm{ ex}_{ q}(n,F)=\big(1-\frac{ 1}{ \chi(F)-1}+o(1) \big)2n$. This solves a problem proposed by Li, Liu and Feng (2022), which gives an entirely satisfactory answer to the problem of estimating $\mathrm{ ex}_q(n,F)$. Furthermore, it extends the aforementioned result of Erd\H{ o}s, Stone and Simonovits as well as the spectral result of Nikiforov. Our result indicates that the Erd\H{ o}s--Stone--Simonovits type result regarding the signless Laplacian spectral radius is valid in general.
Categories: math.CO
