Spectral extremal graphs without intersecting triangles as a minor

Let $F_s$ be the friendship graph obtained from $s$ triangles by sharing a common vertex. For fixed $s\ge 2$ and sufficiently large $n$, the $F_s$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioab\u{a}, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)],and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)]. Recently, the spectral extremal problems was widely studied for graphs containing no $H$ as a minor. For instance, Tait [J. Combin. Theory Ser. A 166 (2019)], Zhai and Lin [J. Combin. Theory Ser. B 157 (2022)] solved the case $H=K_r$ and $H=K_{s,t}$, respectively. Motivated by these results, we consider the spectral extremal problems in the case $H=F_s$. We shall prove that $K_s \vee I_{n-s}$ is the unique graph that attain the maximal spectral radius over all $n$-vertex $F_s$-minor-free graphs. Moreover, let $Q_t$ be the graph obtained from $t$ copies of the cycle of length $4$ by sharing a common vertex. We also determine the unique $Q_t$-minor-free graph attaining the maximal spectral radius. Namely, $K_t \vee M_{n-t}$, where $M_{n-t}$ is a graph obtained from an independent set of order $n-t$ by embedding a matching consisting of $\lfloor \frac{n-t}{2}\rfloor$ edges.