The spectral radius of graphs with no odd wheels

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.