A well-known theorem of Mantel states that every n $n$-vertex graph with more than
& LeftFloor; n 2 / 4 & RightFloor; $\lfloor {n}<^>{2}\unicode{x02215}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 & odblac;s, Faudree, and Rousseau (1992) showed that a graph on n $n$
vertices with more than & LeftFloor; n 2 / 4 & RightFloor; $\lfloor {n}<^>{2}\unicode{x02215}4\rfloor
$ edges contains at least 2 & LeftFloor; n / 2 & RightFloor; + 1 $2\lfloor n\unicode{x02215}2\rfloor
+1$ edges in triangles. Such edges are called triangular edges. In this paper, we
present a spectral version of the result of Erd & odblac;s, Faudree, and Rousseau.
Using the supersaturation-stability and the spectral technique, we prove that every
n $n$-vertex graph G $G$ with lambda ( G ) >= & LeftFloor; n 2 / 4 & RightFloor; $\lambda
(G)\ge \sqrt{\lfloor {n}<^>{2}\unicode{x02215}4\rfloor }$ contains at least 2 & LeftFloor;
n / 2 & RightFloor; - 1 $2\lfloor n\unicode{x02215}2\rfloor -1$ triangular edges,
unless G $G$ 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 & odblac;s concerning the booksize of a graph, which was initially
proved by Edwards (unpublished), and independently by Khad & zcaron;iivanov and Nikiforov
(1979). Secondly, our method can improve the bound on the order n $n$ of the spectral
extremal graph when we forbid the friendship graph as a substructure. We drop the
condition that requires the order n $n$ to be sufficiently large, which was investigated
by Cioab & abreve; 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).
