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

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 Turán numbers.