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

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 $k\ge 2$ is a graph obtained from a cycle of order $2k$ by adding a new vertex connecting to all the vertices of the cycle. Let $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,W_{2k+1})$ be the set of $W_{2k+1}$-free graphs of order $n$ with the maximum spectral radius. Very recently, Cioabă, Desai and Tait \cite{CDT2} characterized the graphs in $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,W_{2k+1})$ for sufficiently large $n$, where $k\ge 2$ and $k\mathrm{\ne }4,5$. 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 $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,W_{2k+1})$ when $k\ge 4$ is even and $n\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$ is sufficiently large. Consequently, the graphs in $\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{X}(n,W_{2k+1})$ are characterized completely for any $k\ge 2$ and sufficiently large $n$.