Spectral supersaturation: Triangles and bowties

A classical result of Erdős and Rademacher (1955) demonstrates a fundamental supersaturation phenomenon in extremal combinatorics: every graph on $n$ vertices with more than $\lfloor n^2/4\rfloor$ edges contains at least $\lfloor n/2\rfloor$ triangles. Let $\lambda \left(G\right)$ be the spectral radius of the adjacency matrix of a graph $G$. Recently, Ning and Zhai (2023) proved that every $n$-vertex graph $G$ with $\lambda \left(G\right)\ge \sqrt{\lfloor n^2/4\rfloor }$ contains at least $\lfloor n/2\rfloor -1$ triangles, unless $G$ is a balanced complete bipartite graph $K_{\lceil \frac{n}{2}\rceil ,\lfloor \frac{n}{2}\rfloor }$. The aim of this paper is two-fold. Using a different approach which we term the supersaturation-stability method, we prove a stability variant of the Ning–Zhai result by showing that such a graph $G$ contains at least $n-3$ triangles if no vertex lies in all triangles of $G$. This bound is the best possible and it could also be viewed as a spectral analogue of a theorem of Xiao and Katona (2021), which guarantees $n-2$ triangles under the assumption that $e\left(G\right)>\lfloor n^2/4\rfloor$ and no vertex is in all triangles of $G$.

The second part concerns with the spectral supersaturation for the bowtie, which consists of two triangles sharing a vertex. Erdős, Füredi, Gould and Gunderson (1995) proved that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor +1$ edges contains a bowtie. The spectral supersaturation problem has not been investigated for non-color-critical substructures in graphs with given order. We give the first such result by counting 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 in the vertex part of size $\lceil \frac{n}{2}\rceil$. We show that if $n\ge 8.8\times 10^6$ and $\lambda \left(G\right)\ge \lambda \left(K_{\lceil \frac{n}{2}\rceil ,\lfloor \frac{n}{2}\rfloor }^{+2}\right)$, then $G$ has 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 result of Kang, Makai and Pikhurko (2020). The method developed in our paper could be helpful in establishing the spectral results for counting other substructures, even for non-color-critical graphs.