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

For a graph G, the spectral radius ${\lambda }_1\left(G\right)$ 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 $\mathrm{SPEX}(n,W_{2k+1})$ be the set of $W_{2k+1}$-free graphs of order n with the maximum spectral radius. Very recently, Cioabă, Desai and Tait [4] characterized the graphs in $\mathrm{SPEX}(n,W_{2k+1})$ for sufficiently large n, where $k\ge 2$ and $k\ne 4,5$. 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 $\mathrm{SPEX}(n,W_{2k+1})$ when $k\ge 4$ is even and $n\equiv 2\left(\text{mod}4\right)$ is sufficiently large. Consequently, the graphs in $\mathrm{SPEX}(n,W_{2k+1})$ are characterized completely for any $k\ge 2$ and sufficiently large n.