Counting substructures and eigenvalues II: quadrilaterals

TL;DR

This work studies the supersaturation of 4-cycles in graphs under the spectral radius condition $\lambda(G)>\,\sqrt{m}$, building on Nikiforov's edge-deletion technique and recent supersaturation results. It proves that the minimum number of copies of $4$-cycles satisfies $f(m)=\Theta(m^2)$, advancing the understanding of how large eigenvalues enforce many occurrences of a fixed subgraph. Central to the argument are a key lemma that leverages Perron-vector products on edges and a deleting-small-eigenvalue-edge (DSEE) method that generates a sequence of graphs while tracking the spectral radius. The results imply strong quadratic growth in $f(m)$ and invite further study of the exact limit and related extremal configurations, with potential implications for spectral extremal theory and probabilistic graph models.

Abstract

Let $G$ be a graph and $λ(G)$ be the spectral radius of $G$. A previous result due to Nikiforov [Linear Algebra Appl., 2009] in spectral graph theory asserted that every graph $G$ on $m\geq 10$ edges contains a 4-cycle if $λ(G)>\sqrt{m}$. Define $f(m)$ to be the minimum number of copies of 4-cycles in such a graph. A consequence of a recent theorem due to Zhai et al. [European J. Combin., 2021] shows that $f(m)=Ω(m)$. In this article, by somewhat different techniques, we prove that $f(m)=Θ(m^2)$. We left the solution to $\lim\limits_{m\rightarrow \infty} \frac{f(m)}{m^2}$ as a problem, and also mention other ones for further study.

Theorems & Definitions (34)

• Theorem 1: N09
• Theorem 2: ZLS21
• Theorem 3
• Proposition 1
• proof
• Lemma 2.1: BH12
• Lemma 2.2: H88
• Lemma 2.3: LL09
• Lemma 2.4: N70BFP08N17
• Lemma 2.5