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Curvature and local matchings of conference graphs and extensions

Kaizhe Chen, Shiping Liu, Heng Zhang

TL;DR

The paper resolves the Bonini et al. conjecture on the Lin--Lu--Yau curvature for conference graphs by deriving parameter-based conditions that force a local perfect matching between core neighborhoods, thereby attaining the curvature upper bound. It introduces a novel combinatorial framework leveraging counts of common neighbors and quadratic-polynomial inequalities to apply Hall's theorem effectively. The method extends beyond conference graphs to broader amply regular graphs and yields a number-theoretic corollary about quadratic residues, notably in Paley-graph contexts. Overall, the work links discrete Ricci curvature upper bounds to local matching structures, advancing understanding of curvature in highly regular graphs.

Abstract

We prove a conjecture of Bonini et al. on the precise values of the Lin--Lu--Yau curvature of conference graphs, i.e., strongly regular graphs with parameters $(4γ+1,2γ,γ-1,γ)$. Our method depends only on the parameter relations and applies to broader classes of amply regular graphs. In particular, we develop a new combinatorial approach to show the existence of local perfect matchings. A key observation is that counting common neighbors leads to useful quadratic polynomials. As a corollary, we derive an interesting number-theoretic result concerning quadratic residues.

Curvature and local matchings of conference graphs and extensions

TL;DR

The paper resolves the Bonini et al. conjecture on the Lin--Lu--Yau curvature for conference graphs by deriving parameter-based conditions that force a local perfect matching between core neighborhoods, thereby attaining the curvature upper bound. It introduces a novel combinatorial framework leveraging counts of common neighbors and quadratic-polynomial inequalities to apply Hall's theorem effectively. The method extends beyond conference graphs to broader amply regular graphs and yields a number-theoretic corollary about quadratic residues, notably in Paley-graph contexts. Overall, the work links discrete Ricci curvature upper bounds to local matching structures, advancing understanding of curvature in highly regular graphs.

Abstract

We prove a conjecture of Bonini et al. on the precise values of the Lin--Lu--Yau curvature of conference graphs, i.e., strongly regular graphs with parameters . Our method depends only on the parameter relations and applies to broader classes of amply regular graphs. In particular, we develop a new combinatorial approach to show the existence of local perfect matchings. A key observation is that counting common neighbors leads to useful quadratic polynomials. As a corollary, we derive an interesting number-theoretic result concerning quadratic residues.
Paper Structure (5 sections, 10 theorems, 63 equations, 1 figure)

This paper contains 5 sections, 10 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lin--Lu--Yau curvature and local matchings
  4. Strongly/Amply regular graph
  5. Proof of Theorem \ref{['thm2']}, Theorem \ref{['Bonini']}, and Corollary \ref{['cor:number_theoretic']}

Key Result

Theorem 1.1

CKKLMP20 Let $G=(V,E)$ be a $d$-regular graph. For any edge $xy\in E$, we have where $\Delta_{xy}:=\Gamma(x)\cap \Gamma(y)$, $\Gamma(x):=\{z\in V| xz\in E\}$, and $\Gamma(y):=\{z\in V| yz\in E\}$. Moreover, the following two propositions are equivalent:

Figures (1)

  • Figure 1: A schematic plot for the decomposition of $V$

Theorems & Definitions (24)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem \ref{thm2}'
  • Corollary 1.5
  • Definition 2.1: Wasserstein distance
  • Definition 2.2: $p$-Ollivier curvature Ollivier09 and Lin--Lu--Yau curvature LLY11
  • Definition 2.3
  • Theorem 2.4
  • ...and 14 more