Spectral extrema of graphs forbidding a fan

TL;DR

This paper characterizes the extremal graphs with maximum spectral radius that do not contain a fan graph as a subgraph, revealing a nested structure between different fan sizes for large graphs.

Contribution

It provides a complete characterization of extremal graphs avoiding fan subgraphs for all large n and all fan sizes, uncovering a new inclusion phenomenon among these extremal sets.

Findings

Characterization of extremal graphs for fan-free graphs

Nested inclusion of extremal sets for different fan sizes

Structural insights into spectral extremal problems

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 $\ell\geq4$. 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 $n$.