Spectral extrema of graphs with bounded clique number and matching number

For a set of graphs $\mathcal{F}$, let $\ex(n,\mathcal{F})$ and $\spex(n,\mathcal{F})$ denote the maximum number of edges and the maximum spectral radius of an $n$-vertex $\mathcal{F}$-free graph, respectively. Nikiforov ({\em LAA}, 2007) gave the spectral version of the Tur\'an Theorem by showing that $\spex(n, K_{k+1})=\lambda (T_{k}(n))$, where $T_k(n)$ is the $k$-partite Tur\'an graph on $n$ vertices. In the same year, Feng, Yu and Zhang ({\em LAA}) determined the exact value of $\spex(n, M_{s+1})$, where $M_{s+1}$ is a matching with $s+1$ edges. Recently, Alon and Frankl~(arXiv2210.15076) gave the exact value of $\ex(n,\{K_{k+1},M_{s+1}\})$. In this article, we give the spectral version of the result of Alon and Frankl by determining the exact value of $\spex(n,\{K_{k+1},M_{s+1}\})$ when $n$ is large.
Comment: 11 pages. arXiv admin note: text overlap with arXiv:2301.05625