A spectral Lov\'{a}sz-Simonovits theorem

TL;DR

This paper establishes a spectral version of a classical extremal graph theory result, linking spectral radius to triangle counts, and identifies extremal graphs with maximum spectral radius under certain conditions.

Contribution

It introduces a spectral analogue of Lovász and Simonovits's supersaturation theorem, providing bounds on spectral radius that guarantee triangle counts and characterizing extremal graphs.

Findings

Spectral radius threshold implies minimum number of triangles.

Bound on q is tight up to a constant factor.

Identifies unique extremal graphs with maximum spectral radius.

Abstract

A fundamental result in extremal graph theory is attributed to Mantel's theorem, which states that every graph on $n$ vertices with more than $\lfloor n^2/4 \rfloor$ edges must contain a triangle. Lov\'{a}sz and Simonovits (1975) provided a supersaturation phenomenon by showing that for any $q< n/2$, every graph with $\lfloor n^2/4 \rfloor +q$ edges contains at least $q\lfloor n/2 \rfloor$ triangles. This result resolved a conjecture proposed by Erd\H{o}s in 1962. In this paper, we establish a spectral counterpart of the result of Lov\'{a}sz and Simonovits. Let $Y_{n,2,q}$ be the graph obtained from the bipartite Tur\'{a}n graph $T_{n,2}$ by embedding a matching with $q$ edges into the partite set of size $\lceil n/2\rceil$. Using the supersaturation-stability method and the spectral techniques, we firstly prove that for $q\le \frac{1}{11}\sqrt{n}$, every graph $G$ on $n$ vertices with spectral radius $\lambda (G) \ge \lambda (Y_{n,2,q})$ contains at least $q\lfloor n/2 \rfloor$ triangles. We also show that the bound $q=O(\sqrt{n})$ is tight up to a constant factor, yielding a phenomenon different from that in edge supersaturation. Our result answers a spectral triangle counting problem proposed by Ning and Zhai (2023). Secondly, let $T_{n,2,q}$ be the graph obtained from $T_{n,2}$ by embedding a star with $q$ edges into the partite set of size $\lceil n/2\rceil$. We show further that $T_{n,2,q}$ is the unique extremal graph that contains at most $q\lfloor n/2 \rfloor$ triangles and attains the maximum spectral radius. Thirdly, we present an asymptotic spectral stability result under a specific constraint on the triangle covering number. This result could be viewed as a spectral extension of a recent result proved by Balogh and Clemen (2023), and independently by Liu and Mubayi (2022).