Extremal problems for disjoint graphs

TL;DR

This paper investigates the relationship between extremal graphs maximizing edges and spectral radius in graphs avoiding multiple disjoint copies of a fixed graph, generalizing previous results and characterizing extremal structures.

Contribution

It proves that for large graphs, the spectral extremal graphs are contained within the edge extremal graphs for multiple disjoint copies, extending prior work.

Findings

Spectral extremal graphs are contained within edge extremal graphs for large n.

Characterization of extremal graphs for multiple disjoint copies of F.

Generalization of previous results on extremal graph theory.

Abstract

For a simple graph $F$, let $\mathrm{EX}(n, F)$ and $\mathrm{EX_{sp}}(n,F)$ be the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph $F$, respectively. Let $F$ be a graph with $\mathrm{ex}(n,F)=e(T_{n,r})+O(1)$. In this paper, we show that $\mathrm{EX_{sp}}(n,kF)\subseteq \mathrm{EX}(n,kF)$ for sufficiently large $n$. This generalizes a result of Wang, Kang and Xue [J. Comb. Theory, Ser. B, 159(2023) 20-41]. We also determine the extremal graphs of $kF$ in term of the extremal graphs of $F$.