An Extremal Problem Motivated by Triangle-Free Strongly Regular Graphs

TL;DR

This paper investigates a combinatorial problem related to triangle-free regular graphs, establishing bounds on the density of common neighborhoods between non-adjacent vertices, with implications for extremal graph configurations.

Contribution

The paper provides new combinatorial bounds on the function a(ρ) for triangle-free regular graphs, identifying extremal configurations and connecting to Krein conditions.

Findings

Derived tight bounds for specific edge densities ρ.

Identified extremal graphs: C_5, Clebsch, Petersen, Higman-Sims.

Connected bounds to Krein conditions for small ρ.

Abstract

We introduce the following combinatorial problem. Let $G$ be a triangle-free regular graph with edge density $\rho$. What is the minimum value $a(\rho)$ for which there always exist two non-adjacent vertices such that the density of their common neighborhood is $\leq a(\rho)$? We prove a variety of upper bounds on the function $a(\rho)$ that are tight for the values $\rho=2/5,\ 5/16,\ 3/10,\ 11/50$, with $C_5$, Clebsch, Petersen and Higman-Sims being respective extremal configurations. Our proofs are entirely combinatorial and are largely based on counting densities in the style of flag algebras. For small values of $\rho$, our bound attaches a combinatorial meaning to Krein conditions that might be interesting in its own right. We also prove that for any $\epsilon>0$ there are only finitely many values of $\rho$ with $a(\rho)\geq\epsilon$ but this finiteness result is somewhat purely existential (the bound is double exponential in $1/\epsilon$).