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

TL;DR

This paper investigates the relationship between Turán extremal graphs and signless Laplacian spectral extremal graphs, establishing conditions under which they coincide for large graphs and specific forbidden subgraphs.

Contribution

It proves that for graphs with chromatic number r+1 and certain Turán number conditions, the extremal graphs for the signless Laplacian spectral radius are contained within the classical Turán extremal graphs.

Findings

For large n, signless Laplacian extremal graphs are subsets of Turán extremal graphs.

The result applies when the Turán number matches the Turán graph plus a bounded error.

The proof employs regularity method and F"{u}redi's stability theorem.

Abstract

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 $T_r(n)$.