Spectral extrema of graphs forbidding a fan

For a graph $G$, its spectral radius is the largest eigenvalue of its adjacency matrix. A fan $H_{\ell}$ is a graph obtained by connecting a single vertex to all vertices of a path of order $\ell\geq4$. Let ${\rm SPEX(n,H_{\ell})}$ be the set of all extremal graphs $G$ of order $n$ with the maximum spectral radius, where $G$ contains no $H_{\ell}$ as a subgraph. In this paper, we completely characterized the graphs in ${\rm SPEX(n,H_{\ell})}$ for any $\ell\geq4$ and sufficiently large $n$. An interesting phenomenon was revealed: ${\rm SPEX(n,H_{2k+2})}\subseteq {\rm SPEX(n,H_{2k+3})}$ for any $k\geq1$ and sufficiently large $n$.