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Finite projective planes meet spectral gaps

Yuhan Guo, Dong Zhang

TL;DR

This work identifies incidence graphs of finite projective planes as the unique extremal structures for spectral-gap problems on graphs. It proves sharp bounds for the adjacency-gap from 0 in terms of maximum degree $d$, achieving equality precisely for incidence graphs of order $d-1$, and improves the bound to $\sqrt{d-2}$ in non-extremal cases; a parallel result for the normalized Laplacian gives a gap from 1 of at most $\sqrt{d-1}/d$, with equality again tied to incidence graphs. A refined bound for graphs with girth at least 7 yields $\sqrt{d-c(d)}/d$ where $c(d)\ge2$ and $c(d)/d\to(\sqrt{5}-1)/2$, showcasing a nontrivial improvement for high-degree, large-girth graphs. The proofs combine a nonregular-to-regular reduction with a detailed analysis of neighborhood graphs $\phi(G)$ and 4-cycle-free structures to pin down the extremal graphs. The results connect spectral-gap extremality to finite geometry and design theory, and they have implications for convergence rates in Laplacian-related processes on graphs.

Abstract

We show that for any connected graph $G$ with maximum degree $d\ge3$, the spectral gap from $0$ with respect to the adjacency matrix is at most $\sqrt{d-1}$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$; and for other cases, the bound $\sqrt{d-1}$ is improved to $\sqrt{d-2}$. This is a spectral gap version of a result by Mohar and Tayfeh-Rezaie. Moreover, for $d$-regular graphs with girth at least 7, the bound $\sqrt{d-2}$ is further improved to $\sqrt{d-c(d)}$ where $c(d)\ge 2$ and $\lim\limits_{d\to\infty}c(d)/d=(\sqrt{5}-1)/2$. A similar yet more subtle phenomenon involving the normalized Laplacian is also investigated, where we work on graphs of degrees $\ge d$ rather than $\le d$. We prove that for any graph $G$ with \emph{minimum} degree $d\ge 3$, the spectral gap from the value 1 with respect to the normalized Laplacian is at most $\sqrt{d-1}/d$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$. As an application, we provide a new sharp bound for the convergence rate of some eigenvalues of the Laplacian on the weighted neighborhood graphs introduced by Bauer and Jost.

Finite projective planes meet spectral gaps

TL;DR

This work identifies incidence graphs of finite projective planes as the unique extremal structures for spectral-gap problems on graphs. It proves sharp bounds for the adjacency-gap from 0 in terms of maximum degree , achieving equality precisely for incidence graphs of order , and improves the bound to in non-extremal cases; a parallel result for the normalized Laplacian gives a gap from 1 of at most , with equality again tied to incidence graphs. A refined bound for graphs with girth at least 7 yields where and , showcasing a nontrivial improvement for high-degree, large-girth graphs. The proofs combine a nonregular-to-regular reduction with a detailed analysis of neighborhood graphs and 4-cycle-free structures to pin down the extremal graphs. The results connect spectral-gap extremality to finite geometry and design theory, and they have implications for convergence rates in Laplacian-related processes on graphs.

Abstract

We show that for any connected graph with maximum degree , the spectral gap from with respect to the adjacency matrix is at most , with equality if and only if is the incidence graph of a finite projective plane of order ; and for other cases, the bound is improved to . This is a spectral gap version of a result by Mohar and Tayfeh-Rezaie. Moreover, for -regular graphs with girth at least 7, the bound is further improved to where and . A similar yet more subtle phenomenon involving the normalized Laplacian is also investigated, where we work on graphs of degrees rather than . We prove that for any graph with \emph{minimum} degree , the spectral gap from the value 1 with respect to the normalized Laplacian is at most , with equality if and only if is the incidence graph of a finite projective plane of order . As an application, we provide a new sharp bound for the convergence rate of some eigenvalues of the Laplacian on the weighted neighborhood graphs introduced by Bauer and Jost.

Paper Structure

This paper contains 9 sections, 19 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary and Main results
  3. Finite projective planes and their incidence graphs
  4. Normalized Laplacian and main results
  5. Proof of the main results
  6. Proof of Theorem \ref{['thm:d-graph-projective']} and Theorem \ref{['th:main2']}
  7. Proof of Theorem \ref{['th:main']}
  8. Proof of Proposition \ref{['thm:girth7']}
  9. Discussions and open problems

Key Result

Theorem 1

Given $d\ge 3$, for any connected graph $G$ with maximum degree $\le d$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$. Furthermore, if a connected graph $G$ has maximum degree $\le d$, and is not the incidence graph of a finite projective plane, then

Figures (3)

  • Figure 1: The Heawood graph
  • Figure 2: The friendship graphs and book graphs used in Theorem \ref{['thm:d=2']}
  • Figure 3: The vertices in the proof of Lemma \ref{['lemma:reduce-to-regular']}

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Definition 1
  • Proposition 2: Godsil
  • Theorem 4
  • Theorem 5
  • Lemma 3
  • proof
  • ...and 23 more