A spectral Erdos-Faudree-Rousseau theorem

A well-known theorem of Mantel states that every n-vertex graph with more than n²/4 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. Erdos, Faudree and Rousseau (1992) showed that a graph on n vertices with more than n²/4 edges contains at least 2 n/2 +1 edges in triangles. Such edges are called triangular edges. In this paper, we present a spectral version of the result of Erdos, Faudree and Rousseau. Using the supersaturation-stability and the spectral technique, we prove that every n-vertex graph G with (G) n²/4 contains at least 2 n/2 -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used to revisit a conjecture of Erdos concerning with the booksize of a graph, which was initially proved by Edwards (unpublished), and independently by Khadziivanov and Nikiforov (1979). Secondly, our method can improve the bound on the order n of a graph by dropping the condition on n being sufficiently large, which is obtained from the triangle removal lemma. Thirdly, the supersaturation-stability can be applied to deal with the spectral extremal graph problems on counting triangles, which was recently studied by Ning and Zhai (2023).