Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral characterizations of local structures of graphs and hypergraphs

Jiang Zhou, Changjiang Bu

TL;DR

The paper investigates how large spectral radius ratios for graphs and hypergraphs enforce specific local subgraph structures. By combining tensor eigenvalue analysis, Lagrangian methods, and theta-function bounds, it derives conditions under which local subhypergraphs and subgraph configurations must appear, such as K1*F and book-like subgraphs, when beta(H) or gamma_r(G) are large. It also connects these spectral quantities to local coloring through the local vector chromatic number, providing bounds that tie tensor-based spectral data to coloring parameters. Overall, the work provides a framework linking global spectral properties to localized structural and coloring consequences in both graphs and hypergraphs.

Abstract

In this paper, we give the relationship between spectral radius and local structures of graphs and hypergraphs. Our work shows that certain local subgraphs (subhypergraphs) must occur when the spectral radius ratio is large. We also give spectral bounds on the local vector chromatic number in terms of tensor eigenvalues of graphs.

Spectral characterizations of local structures of graphs and hypergraphs

TL;DR

The paper investigates how large spectral radius ratios for graphs and hypergraphs enforce specific local subgraph structures. By combining tensor eigenvalue analysis, Lagrangian methods, and theta-function bounds, it derives conditions under which local subhypergraphs and subgraph configurations must appear, such as K1*F and book-like subgraphs, when beta(H) or gamma_r(G) are large. It also connects these spectral quantities to local coloring through the local vector chromatic number, providing bounds that tie tensor-based spectral data to coloring parameters. Overall, the work provides a framework linking global spectral properties to localized structural and coloring consequences in both graphs and hypergraphs.

Abstract

In this paper, we give the relationship between spectral radius and local structures of graphs and hypergraphs. Our work shows that certain local subgraphs (subhypergraphs) must occur when the spectral radius ratio is large. We also give spectral bounds on the local vector chromatic number in terms of tensor eigenvalues of graphs.

Paper Structure

This paper contains 5 sections, 17 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local structure and spectral radius of hypergraphs
  4. Local structure and $r$-clique spectral radius of graphs
  5. Local chromatic number and graph eigenvalues

Key Result

Lemma 2.1

Qi2013 Suppose that $\mathcal{A}=(a_{i_1\cdots i_m})$ is an order $m$ dimension $n$ symmetric nonnegative tensor, with $m\geq2$. Then the spectral radius of $\mathcal{A}$ is equal to where $\mathcal{A}x^m=x^\top(\mathcal{A}x^{m-1})=\sum_{i_1,\ldots,i_m=1}^{n}a_{i_1\cdots i_m}x_{i_1}\cdots x_{i_m}$.

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Theorem 3.1
  • proof
  • ...and 15 more