Spectral extremal graphs for disjoint odd wheels

Abstract

For a given graph F, let $\mathrm{ex}(n,F)$ and $\mathrm{spex}(n,F)$ be the maximum number of edges and the maximum spectral radius of the adjacency matrix over all F-free graphs of order n, respectively. $\mathrm{EX}(n,F)$ and $\mathrm{SPEX}(n,F)$ consist of the extremal graphs associated with $\mathrm{ex}(n,F)$ and $\mathrm{spex}(n,F)$, respectively. The odd wheel $W_{2k+1}$ is constructed by joining a vertex to a cycle $C_{2k}$. Cioabă, Desai and Tait determined the spectral extremal graphs of $W_{2k+1}$ for $k\ge 2,k\notin \{4,5\}$. Xiao and Zamora determined the Turán number and all extremal graphs for $t{W}_{2k+1}$, which is the union of t vertex-disjoint copies of $W_{2k+1}$ for $k\ge 3$. In this paper, we focus on the graph with maximum spectral radius among those that exclude any subgraph isomorphic to $t{W}_{2k+1}$. We present structural characteristics of these spectral extremal graphs for $k\ge 3,k\notin \{4,5\}$. Furthermore, we demonstrate that $\mathrm{SPEX}(n,t{W}_{2k+1})\cap \mathrm{EX}(n,t{W}_{2k+1})=\varnothing$ for $k\ge 10$ and n large enough.