Turán extremal graphs vs. Signless Laplacian spectral Turán extremal graphs

Ming-Zhu Chen; Ya-Lei Jin; Peng-Li Zhang; Xiao-Dong Zhang

paper

Turán extremal graphs vs. Signless Laplacian spectral Turán extremal graphs

arXiv:2602.11502

Abstract

Let $F$ be a graph with chromatic number $χ (F) = r + 1$ . Denote by $e x (n, F)$ and $E x (n, F)$ the Turán number and the set of all extremal graphs for $F$ , respectively. In addition, $e x_{ss p} (n, F)$ and $E x_{ss p} (n, F)$ 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 $e x_{ss p} (n, F)$ , respectively. It is known that $E x_{ss p} (n, F) \supset E x (n, F)$ if $F$ is a triangle. In this paper, employing the regularity method and Füredi's stability theorem, we prove that for a given graph $F$ and $r ⩾ 3$ , if $e x (n, F) = t_{r} (n) + O (1)$ , then $E x_{ss p} (n, F) \subseteq E x (n, F)$ for sufficiently large $n$ , where $t_{r} (n)$ is the number of edges in the Turán graph $T_{r} (n)$ .

20 pages