The $\mathcal A_α$-Spectral Turán Problem for $s$-Clique Tensors of $K_{r+1}$-Free Graphs

Source abstract

For fixed $r\geq3$ and $0 c_r$. For general $3\leq s\leq r$ and $n\geq r$, every extremal graph is complete $r$-partite. The asymptotic maximum $α$-$r$-clique spectral radius among $n$-vertex $K_{r+1}$-free graphs is $\max\{r^{1-r},α(r-1)^{1-r}\}n^{r-1}+o(n^{r-1})$. In every asymptotically extremal sequence of complete $r$-partite graphs, all parts have size $n/r+o(n)$ when $α c_r$, one part has size $o(n)$ and the remaining parts have size $n/(r-1)+o(n)$. At $α=c_r$, both part-size patterns are asymptotically extremal.