More on spectral supersaturation for the bowtie

TL;DR

This paper investigates the spectral supersaturation problem for the bowtie graph, establishing bounds on the spectral radius that guarantee the presence of multiple bowties in large graphs, thus extending extremal graph theory results.

Contribution

It proves a spectral supersaturation theorem for the bowtie, identifying the unique extremal graph and establishing bounds on bowtie counts based on spectral radius.

Findings

Spectral radius bounds guarantee multiple bowties in large graphs.

Identification of the unique spectral extremal graph for the bowtie.

Sharp bounds on bowtie counts differ from edge-based supersaturation results.

Abstract

A central topic in extremal graph theory is the supersaturation problem, which studies the minimum number of copies of a fixed substructure that must appear in any graph with more edges than the corresponding Tur\'an number. Significant works due to Erd\H{o}s, Rademacher, Lov\'{a}sz and Simonovits investigated the supersaturation problem for the triangle. Moreover, Kang, Makai and Pikhurko studied the case for the bowtie, which consists of two triangles sharing a vertex. Building upon the pivotal results established by Bollob\'{a}s, Nikiforov, Ning and Zhai on counting triangles via the spectral radius, we study in this paper the spectral supersaturation problem for the bowtie. Let $\lambda (G)$ be the spectral radius of a graph $G$, and let $K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q$ be the graph obtained from Tur\'{a}n graph $T_{n,2}$ by adding $q$ pairwise disjoint edges to the partite set of size $\lceil \frac{n}{2}\rceil$. Firstly, we prove that there exists an absolute constant $\delta >0$ such that if $n$ is sufficiently large, $2\le q \le \delta \sqrt{n}$, and $G$ is an $n$-vertex graph with $\lambda (G)\ge \lambda (K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q)$, then $G$ contains at least ${q\choose 2}\lfloor \frac{n}{2}\rfloor$ bowties, and $K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q$ is the unique spectral extremal graph. This solves an open problem proposed by Li, Feng and Peng. Secondly, we show that a graph $G$ whose spectral radius exceeds that of the spectral extremal graph for the bowtie must contain at least $\lfloor \frac{n-1}{2}\rfloor$ bowties. This sharp bound reveals a distinct phenomenon from the edge-supersaturation case, which guarantees at least $\lfloor \frac{n}{2}\rfloor$ bowties.