A sharp spectral extremal result for general non-bipartite graphs

TL;DR

This paper establishes a precise spectral extremal result for non-bipartite graphs, determining the maximum spectral radius for graphs avoiding a certain family, extending previous bounds and answering an open question.

Contribution

The paper exactly determines the maximum constant c(r) for spectral extremal bounds in non-bipartite graphs, generalizing prior approximate results.

Findings

Exact value of c(r) for all r ≥ 3

Extension of spectral extremal results to broader graph families

Resolution of an open problem in spectral graph theory

Abstract

For a graph family $\mathcal F$, let $\mathrm{ex}(n,\mathcal F)$ and $\mathrm{spex}(n,\mathcal F)$ denote the maximum number of edges and maximum spectral radius of an $n$-vertex $\mathcal F$-free graph, respectively, and let $\mathrm{EX}(n,\mathcal F)$ and $\mathrm{SPEX}(n,\mathcal F)$ denote the corresponding sets of extremal graphs. Wang, Kang, and Xue showed that if $r\ge 2$ and $\mathrm{ex}(n,F)=e(T_{n,r})+O(1)$ then $\mathrm{SPEX}(n,\mathcal F)\subseteq\mathrm{EX}(n,\mathcal F)$ for $n$ large enough. Fang, Tait, and Zhai extended this result by showing if $e(T_{n,r})\le\mathrm{ex}(n,\mathcal F)<e(T_{n,r})+\lfloor n/2r\rfloor$ then $\mathrm{SPEX}(n,\mathcal F)\subseteq\mathrm{EX}(n,\mathcal F)$ for $n$ large enough, and asked for the maximum constant $c(r)$ such that $\mathrm{ex}(n,\mathcal F)\le e(T_{n,r})+(c(r)-\varepsilon)n$ guarantees such containment. In this paper we determine $c(r)$ exactly for all $r\ge 3$.