  Recently, Ning and Zhai (2023) proved that every $n$-vertex graph $G$ with
$\lambda (G) \ge \sqrt{\lfloor n^2/4\rfloor}$ has at least $\lfloor n/2\rfloor
-1$ triangles, unless $G=K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2}
\rfloor}$. The aim of this paper is two-fold. Using the
supersaturation-stability method, we prove a stability variant of Ning-Zhai's
result by showing that such a graph $G$ contains at least $n-3$ triangles if no
vertex is in all triangles of $G$. This result could also be viewed as a
spectral version of a result of Xiao and Katona (2021). The second part
concerns with the spectral supersaturation for the bowtie, which consists of
two triangles sharing a common vertex. A theorem of Erd\H{o}s, F\"{u}redi,
Gould and Gunderson (1995) says that every $n$-vertex graph with more than
$\lfloor n^2/4\rfloor +1$ edges contains a bowtie. For graphs of given order,
the spectral supersaturation problem has not been considered for substructures
that are not color-critical. In this paper, we give the first such theorem by
counting the number of bowties. Let $K_{\lceil \frac{n}{2} \rceil, \lfloor
\frac{n}{2} \rfloor}^{+2}$ be the graph obtained from $K_{\lceil \frac{n}{2}
\rceil, \lfloor \frac{n}{2} \rfloor}$ by embedding two disjoint edges into the
vertex part of size $\lceil \frac{n}{2} \rceil$. Our result shows that every
graph $G$ with $n\ge 8.8 \times 10^6$ vertices and $\lambda (G)\ge \lambda
(K_{\lceil \frac{n}{2} \rceil, \lfloor \frac{n}{2} \rfloor}^{+2})$ contains at
least $\lfloor \frac{n}{2} \rfloor$ bowties, and $K_{\lceil \frac{n}{2} \rceil,
\lfloor \frac{n}{2} \rfloor}^{+2}$ is the unique spectral extremal graph. This
gives a spectral correspondence of a theorem of Kang, Makai and Pikhurko
(2020). The method used in our paper provides a probable way to establish the
spectral counting results for other graphs, even for non-color-critical graphs.
