The high order spectral extrema of $2K_r$-free graphs

Abstract

In this paper, we determine the graphs with maximum value of the sum number from $k$-clique spectral radius to $(2r-1)$-clique spectral radius among all $2K_{r}$-free graphs on $n$ vertices for $ r\le k$ and large $n$. We also determine the graphs with maximum $3$-clique spectral radius among all $2K_{3}$-free graphs on $n$ vertices. Our results are spectral versions of some results on generalized Tur\'an numbers.