  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 $n$.
