Spectral extremal problems for graphs with bounded clique number

For a family of graphs $F$, a graph is called $F$-free if it contains no subgraph isomorphic to any graph in $F$. For two integers n and r, let $G(n,r)$ be the set of graphs on n vertices with clique number at most r. Denote by${\text{ex}}_{sp}(n,r,F)=\max \left\{\lambda \right(G\left)\right|G\in G(n,r)\text{and}G\text{is }F\text{-free}\},$ where $\lambda \left(G\right)$ is the spectral radius of G. Furthermore, denote by ${\text{Ex}}_{sp}(n,r,F)=\{G\in G(n,r\left)\right|\lambda \left(G\right)={\text{ex}}_{sp}(n,r,F),G\text{ is }F\text{-}\text{free}\}$ the set of extremal graphs. In this paper, we first give a spectral Erdös-Sós theorem in $G(n,r)$, that is, for fixed $k\ge r\ge 2$ and sufficiently large n, if a graph $G\in G(n,r)$ with $\lambda \left(G\right)\ge \lambda \left(T_{r-1}\right(k)\vee I_{n-k})$, then it contains all trees on $2k+2$ vertices or $2k+3$ vertices unless $G=T_{r-1}\left(k\right)\vee I_{n-k}$, where $T_{r-1}\left(k\right)\vee I_{n-k}$ is the join of the Turán graph $T_{r-1}\left(k\right)$ and the independent set $I_{n-k}$. Next, for fixed $k\ge r\ge 2$ and sufficiently large n, we completely determine ${\text{Ex}}_{sp}(n,r,C_{2k+2})$, that is ${\text{Ex}}_{sp}(n,r,C_{2k+2})=\left\{T_{r-1}\right(k)\vee I_{n-k}\}$.