More on spectral supersaturation for the bowtie

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.