Generalized spectral Tur\'an problems for disjoint cliques

Abstract

The generalized Tur\'an number $\text{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Let $kK_{r+1}$ be the disjoint union of $k$ copies of the complete graph $K_{r+1}$. Recently, Gerbner determined $\text{ex}(n, K_{t},kK_{r+1})$ 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 $kK_{r+1}$-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 Tur\'an 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 $kK_{r+1}$-free graphs.