More results on the spectral radius of graphs with no odd wheels

TL;DR

This paper completes the characterization of graphs with maximum spectral radius that do not contain odd wheels, resolving previously open cases for specific wheel sizes and large graph orders.

Contribution

It fully characterizes the extremal graphs avoiding odd wheels for all sizes and sufficiently large orders, including previously unresolved cases.

Findings

Resolved the case for $k=4,5$ in odd wheel-free graphs.

Characterized extremal graphs for even $k eq4,5$ when $n ot rsim 4$.

Provided complete characterization for all $k eq2$ and large $n$.

Abstract

For a graph $G$, the spectral radius $\lambda_{1}(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. An odd wheel $W_{2k+1}$ with $k\geq2$ is a graph obtained from a cycle of order $2k$ by adding a new vertex connecting to all the vertices of the cycle. Let ${\rm SPEX}(n,W_{2k+1})$ be the set of $W_{2k+1}$-free graphs of order $n$ with the maximum spectral radius. Very recently, Cioab\u{a}, Desai and Tait \cite{CDT2} characterized the graphs in ${\rm SPEX}(n,W_{2k+1})$ for sufficiently large $n$, where $k\geq2$ and $k\neq4,5$. And they left the case $k=4,5$ as a problem. In this paper, we settle this problem. Moreover, we completely characterize the graphs in ${\rm SPEX}(n,W_{2k+1})$ when $k\geq4$ is even and $n\equiv2~(\mod4)$ is sufficiently large. Consequently, the graphs in ${\rm SPEX}(n,W_{2k+1})$ are characterized completely for any $k\geq2$ and sufficiently large $n$.