Spectral extremal problem of the $p$th power of cycles

For a cycle $C_k$ on $k$ vertices, its $p$-th power, denoted $C_k^p$, is the graph obtained by adding edges between all pairs of vertices at distance at most $p$ in $C_k$. Let $\text{ex}(n,F)$ and $\text{spex}(n,F)$ denote the maximum possible number of edges and the maximum possible spectral radius, respectively, among all $n$-vertex $F$-free graphs. In this paper, we determine precisely the unique extremal graph achieving $\text{ex}(n,C_k^p)$ and $\text{spex}(n,C_k^p)$ for sufficiently large $n$.