  Let $G$ be a graph and $\lambda(G)$ be the spectral radius of $G$. A previous
result due to Nikiforov [Linear Algebra Appl., 2009] in spectral graph theory
asserted that every graph $G$ on $m\geq 10$ edges contains a 4-cycle if
$\lambda(G)>\sqrt{m}$. Define $f(m)$ to be the minimum number of copies of
4-cycles in such a graph. A consequence of a recent theorem due to Zhai et al.
[European J. Combin., 2021] shows that $f(m)=\Omega(m)$. In this article, by
somewhat different techniques, we prove that $f(m)=\Theta(m^2)$. We left the
solution to $\lim\limits_{m\rightarrow \infty} \frac{f(m)}{m^2}$ as a problem,
and also mention other ones for further study.
