Turán extremal graphs vs. Signless Laplacian spectral Turán extremal graphs

Let $F$ be a graph with chromatic number $χ(F) = r+1$. Denote by $ex(n, F)$ and $Ex(n, F)$ the Turán 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 Füredi’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 Turán graph $T_r(n)$.

💡 Research Summary

The paper investigates the relationship between two classical extremal problems in graph theory: the Turán extremal problem, which asks for the maximum number of edges in an n‑vertex F‑free graph, and its spectral analogue involving the signless Laplacian matrix. For a graph F with chromatic number χ(F)=r+1, the Turán number ex(n,F) and the family of extremal graphs Ex(n,F) are well‑studied. In the spectral setting, ex_ssp(n,F) denotes the largest signless Laplacian spectral radius among all n‑vertex F‑free graphs, and Ex_ssp(n,F) is the collection of graphs attaining this maximum. It is known that for r=2 (e.g., triangles, odd cycles, books) the two families can differ dramatically: Ex_ssp may contain many non‑Turán graphs, and sometimes the intersection is empty.

The authors focus on the case r ≥ 3 and assume that the Turán number is essentially the Turán graph value, i.e., ex(n,F)=t_r(n)+O(1), where t_r(n) is the edge count of the balanced complete r‑partite graph T_r(n). Under this assumption they prove that for sufficiently large n the spectral extremal family coincides with the classical one: \