# Spectral Extremal Problem on Disjoint Color-Critical Graphs

### Abstract

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 $K_r^{+}(a_1, a_2, \ldots, a_r)$ be the graph obtained from complete $r$-partite graph with parts of sizes $a_1\geqslant 2, a_2, \ldots, a_r$, by adding an edge to the first part. In this paper, we focus on the spectral extrema of disjoint color-critical graphs. For fixed $t, a_1,\ldots, a_r\ (r\geqslant 2)$ and large enough $n$, we characterize the unique $n$-vertex $tK_r^+(a_1,\ldots,a_r)$-free graph having the largest spectral radius. Moreover, let $F_1,\ldots,F_t$ be $t$ disjoint color-critical graphs with the same chromatic number. We identify the unique $n$-vertex $\bigcup_{i=1}^tF_i$-free graph having the largest spectral radius for sufficiently large $n$. Consequently, we generalize the main results obtained by Ni, Wang and Kang [Electron. J. Combin. 30 (1) (2023), #P1.20] and by Fang, Zhai and Lin [arXiv:2302.03229v2].