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

Abstract

For fixed $r\geq3$ and $0<α<1$, we study the maximum $α$-$s$-clique spectral radius among $n$-vertex $K_{r+1}$-free graphs. Here $\mathcal A_{α,s}(G)=α\mathcal D_s(G)+(1-α)\mathcal A_s(G)$, where $\mathcal A_s(G)$ and $\mathcal D_s(G)$ are the adjacency and degree tensors of the $s$-clique hypergraph of $G$. For $s=r$, put $c_r=((r-1)/r)^{r-1}$. We prove that, for all sufficiently large $n$, the unique extremal graph is the Turán graph $T_r(n)$ if $α\leq c_r$, and $K_1\vee T_{r-1}(n-1)$ if $α>c_r$. 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.