Tur\'{a}n extremal graphs vs. Signless Laplacian spectral Tur\'{a}n extremal graphs

Let $F$ be a graph with chromatic number $\chi(F) = r+1$. Denote by $ex(n, F)$ and $Ex(n, F)$ the Tur\'{a}n number and the set of all extremal graphs for $F$, respectively. In addition, $ex_{ssp}(n, F)$ and $Ex_{ssp}(n, F)$ are the maximum signless Laplacian spectral radius of all $n$-vertex $F$-free graphs and the set of all $n$-vertex $F$-free graphs with signless Laplacian spectral radius $ex_{ssp}(n, F)$, respectively. It is known that $Ex_{ssp}(n, F)\supset Ex(n, F)$ if $F$ is a triangle. In this paper, employing the regularity method and F\"{u}redi's stability theorem, we prove that for a given graph $F$ and $r\geqslant 3$, if $ex(n, F) = t_r(n)+O(1)$, then $Ex_{ssp}(n, F) \subseteq Ex(n, F)$ for sufficiently large $n$, where $t_r(n)$ is the number of edges in the Tur\'{a}n graph $T_r(n)$.