Spectral Extremal Graphs without Intersecting Triangles as a Minor

Abstract

Let $F_s$ be the friendship graph obtained from $s$ triangles by sharing a common vertex. For every $s\ge 2$ and $n\ge 50s^2$, the Turán number of $F_s$ was investigated by Erdős, Füredi, Gould and Gunderson (1995). For sufficiently large $n$,  the $F_s$-free graphs of order $n$ which attain the maximum spectral radius were firstly characterized by Cioabă, Feng, Tait and Zhang (2020), and later uniquely determined by Zhai, Liu and Xue (2022). Recently, the spectral extremal problems were studied for graphs that do not contain a certain graph $H$ as a minor. For instance, Tait (2019), Zhai and Lin (2022), Chen, Liu and Zhang (2024) solved the case of cliques, bicliques, cliques with some paths removed, respectively. Motivated by these results, we consider the spectral extremal problem for friendship graphs. Let $K_s\vee I_{n-s}$ be the complete split graph, which is the join of a clique of size $s$ with an independent set of size $n-s$. For sufficiently large $n$, we prove that $K_s \vee I_{n-s}$ is the unique graph that attains the maximal spectral radius over all $n$-vertex $F_s$-minor-free graphs.