Spectral Extremal Problem on Disjoint Color-Critical Graphs

For a given graph F, we say that a graph G is F -free if it does not contain F as a subgraph. A graph is color -critical if it contains an edge whose deletion reduces its chromatic number. Let Kr+ (a1, a2,... , ar) be the graph obtained from complete r -partite graph with parts of sizes a1 2, a2, ... , ar, by adding an edge to the first part. In this paper, we focus on the spectral extrema of disjoint colorcritical graphs. For fixed t, a1, ... , ar (r 2) and large enough n, we characterize the unique n -vertex tKr+ (a1, ... , ar)-free graph having the largest spectral radius. Moreover, let F1, ... , Ft be t disjoint color -critical graphs with the same chromatic number. We identify the unique n -vertex Uti=1 Fi-free graph having the largest spectral radius for sufficiently large n. Consequently, we generalize the main results obtained by Ni, Wang and Kang