  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.
