Spectral supersaturation: Triangles and bowties

TL;DR

This paper extends spectral supersaturation results to triangles and bowties, providing stability variants and counting the number of such subgraphs in large graphs based on spectral properties.

Contribution

It introduces a spectral stability theorem for triangles and the first spectral supersaturation result for bowties, even for non-color-critical graphs.

Findings

Graphs with high spectral radius contain many triangles or bowties.

Stability results show graphs close to extremal have nearly the minimal number of triangles.

Spectral extremal graphs are characterized for bowties in large graphs.

Abstract

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.