Subdivision-free graphs with the maximum spectral radius

Given a graph family $\mathbb{H}$, let $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,{\mathbb{H}}_{\mathrm{s}\mathrm{u}\mathrm{b}})$ denote the set of $n$-vertex $\mathbb{H}$-subdivision-free graphs with the maximum spectral radius. In this paper, we investigate the problem of graph subdivision from a spectral extremal perspective, with a focus on the structural characterization of graphs in $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,{\mathbb{H}}_{\mathrm{s}\mathrm{u}\mathrm{b}})$. For any graph $H\in \mathbb{H}$, let $\alpha (H)$ denote its independence number. Define ${\gamma }_{\mathbb{H}}:={\min }_{H\in \mathbb{H}}\{\mathrm{\mid }H\mathrm{\mid }-\alpha (H)-1\}$. We prove that every graph in $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,{\mathbb{H}}_{\mathrm{s}\mathrm{u}\mathrm{b}})$ contains a spanning subgraph isomorphic to $K_{{\gamma }_{\mathbb{H}}}\vee (n-{\gamma }_{\mathbb{H}})K_1$, which is obtained by joining a ${\gamma }_{\mathbb{H}}$-clique with an independent set of $n-{\gamma }_{\mathbb{H}}$ vertices. This extends a recent result by Zhai, Fang, and Lin concerning spectral extremal problems for $\mathbb{H}$-minor-free graphs.