A Spectral Erdős–Faudree–Rousseau Theorem

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. Erdős, 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 Erdős, Faudree, and Rousseau. Using the supersaturation-stability and the spectral technique, we prove that every $n$-vertex graph $G$ with $\lambda \left(G\right)\ge \sqrt{\lfloor n^2∕4\rfloor }$ 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can be used to revisit a conjecture of Erdős concerning the booksize of a graph, which was initially proved by Edwards (unpublished), and independently by Khadžiivanov and Nikiforov (1979).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 Cioabă et al. (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, supersaturation stability could be applied to deal with the spectral graph problems on counting triangles, which was recently studied by Ning and Zhai (2023).