Spectral extrema of $\{K_{k+1},{\mathcal{L}}_s\}$-free graphs

For a set of graphs $\mathcal{F}$, a graph is said to be $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. Let Ex${}_{sp}(n,\mathcal{F})$ denote the graphs with the maximum spectral radius among all $\mathcal{F}$-free graphs of order $n$. A linear forest is a graph whose connected component is a path. Denote by ${\mathcal{L}}_s$ the family of all linear forests with $s$ edges. In this paper the graphs in Ex${}_{sp}(n,\{K_{k+1},{\mathcal{L}}_s\})$ will be completely characterized when $n$ is appropriately large.