The spectral radius of graphs with no odd wheels

TL;DR

This paper characterizes the maximum spectral radius of graphs without odd wheels, revealing the structure of extremal graphs and their relation to Turán graphs for various values of k.

Contribution

It determines the spectral extremal graphs avoiding odd wheels for all k≥2, except for k=4,5, and compares them to Turán-extremal graphs.

Findings

Spectral extremal graphs are Turán-extremal for k=2.

For k≥9, spectral extremal graphs differ from Turán-extremal graphs.

The structure of extremal graphs varies with k, with exceptions at k=4,5.

Abstract

The odd wheel $W_{2k+1}$ is the graph formed by joining a vertex to a cycle of length $2k$. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an $n$-vertex graph that does not contain $W_{2k+1}$. We determine the structure of the spectral extremal graphs for all $k\geq 2, k\not\in \{4,5\}$. When $k=2$, we show that these spectral extremal graphs are among the Tur\'{a}n-extremal graphs on $n$ vertices that do not contain $W_{2k+1}$ and have the maximum number of edges, but when $k\geq 9$, we show that the family of spectral extremal graphs and the family of Tur\'{a}n-extremal graphs are disjoint.