Spectral extrema of graphs forbidding a fan

Abstract

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 $\ell \ge 4$. Let $\mathrm{SPEX}(\mathrm{n},{\mathrm{H}}_{\mathrm{\ell }})$ 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 $\mathrm{SPEX}(\mathrm{n},{\mathrm{H}}_{\mathrm{\ell }})$ for any $\ell \ge 4$ and sufficiently large $n$. An interesting phenomenon was revealed: $\mathrm{SPEX}(\mathrm{n},{\mathrm{H}}_{2k+2})\subseteq \mathrm{SPEX}(\mathrm{n},{\mathrm{H}}_{2k+3})$ for any $k\ge 1$ and sufficiently large $n$.