More on spectral supersaturation for the bowtie
Longfei Fang, Yongtao Li, Huiqiu Lin
TL;DR
The paper advances the spectral supersaturation program for the bowtie $F_2$ by establishing a structural framework around graphs with large spectral radius that minimize bowtie copies. It proves a sharp bound: for large $n$ and $2\le q\le\delta\sqrt{n}$, any $n$-vertex graph $G$ with $\lambda(G)\ge\lambda(K_{\lceil n/2\rceil,\lfloor n/2\rfloor}^{q})$ contains at least $\binom{q}{2}\lfloor n/2\rfloor$ bowties, with equality only for the extremal graph $K_{\lceil n/2\rceil,\lfloor n/2\rfloor}^{q}$. If $\lambda(G)$ surpasses the spectral extremal threshold $\lambda(K_{\lceil n/2\rceil,\lfloor n/2\rfloor}^{1})$, the Bowties count is bounded below by $\lfloor( n-1)/2\rfloor$, and the authors provide a detailed stability description of $G$ as a near-bipartite graph with controlled class-edge and cross-edge alterations. The results fully resolve the conjecture in the stated regime and yield a precise extremal/even-odd dichotomy for the structure of $G$, enriching the connection between spectral radius and supersaturation phenomena in non-color-critical settings.
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 Turán number. Significant works due to Erdős, Rademacher, Lovász 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 Bollobás, Nikiforov, Ning and Zhai on counting triangles via the spectral radius, we study in this paper the spectral supersaturation problem for the bowtie. Let $λ(G)$ 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 Turán 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 $δ>0$ such that if $n$ is sufficiently large, $2\le q \le δ\sqrt{n}$, and $G$ is an $n$-vertex graph with $λ(G)\ge λ(K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q)$, 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.