Clique spectral extremal problem on disjoint color-critical graphs

Abstract

For a given graph $F$, a graph $G$ is called $F$-free if it does not contain $F$ as a subgraph. A graph is color-critical if deleting one of its edges decreases its chromatic number. Let $F_1,F_2,\cdots ,F_t$ be $t$ disjoint color-critical graphs with chromatic number $r+1$. For $2\le s\le r$ and sufficiently large $n$, we determine the unique extremal graph with the maximum $s$-clique spectral radius among all $n$-vertex ${\bigcup }_{i=1}^t{F}_i$-free graphs.