Generalized spectral Turán problems for disjoint cliques

Abstract

The generalized Turán number $\text{ex}(n,H,F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Let $k{K}_{r+1}$ be the disjoint union of $k$ copies of the complete graph $K_{r+1}$. Recently, Gerbner determined $\text{ex}(n,K_t,k{K}_{r+1})$ for all sufficiently large $n$. In this paper, we study a spectral analogue of this problem via the $t$-clique tensor of a graph. We prove that if an $n$-vertex $k{K}_{r+1}$-free graph $G$ maximizes the $t$-clique spectral radius, then for sufficiently large $n$, $G$ is the join of a complete graph $K_{k-1}$ and the $r$-partite Turán graph $T_r(n-k+1)$. This establishes a spectral counterpart of Gerbner's Theorem. Moreover, in the case $t=2$, our result recovers a theorem of Ni, Wang, and Kang on the maximum spectral radius of $k{K}_{r+1}$-free graphs.