Some stability results for spectral extremal problems of graphs with bounded matching number

For a set of graphs $H$, a graph is called $H$-free if it does not contain any member of $H$ as a subgraph. The maximum value of spectral radius among all $H$-free graphs of order n is denoted by $\mathrm{spex}(n,H)$, and the set of corresponding extremal graphs is denoted by $\mathrm{SPEX}(n,H)$. In this paper, we give a stability result for graphs in $\mathrm{SPEX}(n,H)$ when $\mathrm{spex}(n,H)\ge \sqrt{s(n-s)}$ and $\mathrm{ex}(n,H)\le sn$. As an application, we may give some characterizations for the graphs in $\mathrm{SPEX}(n,\{M_{s+1},H\left\}\right)$, where $M_{s+1}$ is a matching with $s+1$ edges and H is any non-bipartite graph.