A signless Laplacian spectral Erdös-Stone-Simonovits theorem

The celebrated Erdős–Stone–Simonovits theorem states that $\mathrm{e}\mathrm{x}(n,F)=(1-\frac{1}{\chi (F)-1}+o(1))\frac{n^2}{2}$, where $\chi (F)$ is the chromatic number of $F$. In 2009, Nikiforov proved a spectral extension of the Erdős–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 ${\mathrm{e}\mathrm{x}}_q(n,F)=\max \{q(G):\mathrm{\mid }G\mathrm{\mid }=nandF⊈G\}$. It is known that the Erdős–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 $\chi (F)\ge 3$, then ${\mathrm{e}\mathrm{x}}_q(n,F)=(1-\frac{1}{\chi (F)-1}+o(1))2n$. This solves a problem proposed by Li, Liu and Feng (2022), which gives an entirely satisfactory answer to the problem of estimating ${\mathrm{e}\mathrm{x}}_q(n,F)$. Furthermore, it extends the aforementioned result of Erdős, Stone and Simonovits as well as the spectral result of Nikiforov. Our result indicates that the Erdős–Stone–Simonovits type result regarding the signless Laplacian spectral radius is valid in general.