A spectral Erd\H{o}s-Faudree-Rousseau theorem

TL;DR

This paper extends classical extremal graph results to a spectral setting, showing that graphs with large spectral radius contain many triangular edges unless they are bipartite, with applications to various spectral extremal problems.

Contribution

It introduces a spectral version of Erd ext{"o}s, Faudree, and Rousseau's theorem, using supersaturation-stability and spectral techniques to derive new bounds and applications.

Findings

Graphs with spectral radius above a threshold contain many triangular edges.

The method improves bounds on spectral extremal problems involving forbidden subgraphs.

Applications include revisiting classical conjectures and stability results in spectral graph theory.

Abstract

A well-known theorem of Mantel states that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor $ edges contains a triangle. An interesting problem in extremal graph theory studies the minimum number of edges contained in triangles among graphs with a prescribed number of vertices and edges. Erd\H{o}s, Faudree and Rousseau (1992) showed that a graph on $n$ vertices with more than $\lfloor n^2/4\rfloor $ edges contains at least $2\lfloor n/2\rfloor +1$ edges in triangles. Such edges are called triangular edges. In this paper, we present a spectral version of the result of Erd\H{o}s, Faudree and Rousseau. Using the supersaturation-stability and the spectral technique, we prove that every $n$-vertex graph $G$ with $\lambda (G) \ge \sqrt{\lfloor n^2/4\rfloor}$ contains at least $2 \lfloor {n}/{2} \rfloor -1$ triangular edges, unless $G$ is a balanced complete bipartite graph. The method in our paper has some interesting applications. Firstly, the supersaturation-stability can be used to revisit a conjecture of Erd\H{o}s concerning with the booksize of a graph, which was initially proved by Edwards (unpublished), and independently by Khad\v{z}iivanov and Nikiforov (1979). Secondly, our method can improve the bound on the order $n$ of the spectral extremal graph when we forbid the friendship graph as a substructure. We drop the condition that requires the order $n$ to be sufficiently large, which was investigated by Cioab\u{a}, Feng, Tait and Zhang (2020) using the triangle removal lemma. Thirdly, this method can be utilized to deduce the classical stability for odd cycles and it gives more concise bounds on parameters. Finally, the supersaturation-stability could be applied to deal with the spectral graph problems on counting triangles, which was recently studied by Ning and Zhai (2023).