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Spectral bounds for the independence number of graphs and even uniform hypergraphs

Xinyu Hu, Jiang Zhou, Changjiang Bu

Abstract

In this paper, we give spectral upper bounds for the independence number of even uniform hypergraphs and graphs, extend the Hoffman bound to even uniform hypergraphs, and give a simple spectral condition for determining the independence number, the Shannon capacity and the Lovász number of a graph. The Hoffman bound on the Lovász number is also extended from regular graphs to general graphs.

Spectral bounds for the independence number of graphs and even uniform hypergraphs

Abstract

In this paper, we give spectral upper bounds for the independence number of even uniform hypergraphs and graphs, extend the Hoffman bound to even uniform hypergraphs, and give a simple spectral condition for determining the independence number, the Shannon capacity and the Lovász number of a graph. The Hoffman bound on the Lovász number is also extended from regular graphs to general graphs.
Paper Structure (6 sections, 11 theorems, 60 equations)

This paper contains 6 sections, 11 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lovász number
  4. Spectrum of hypergraphs
  5. Hoffman bound for even uniform hypergraphs
  6. Independence number and Lovász number of graphs

Key Result

Theorem 1.1

Brouwer Let $G$ be an $n$-vertex $d$-regular ($d>0$) graph with minimum eigenvalue $\lambda$. Then If an independent set $S$ meets this bound, then every vertex not in $S$ is adjacent to exactly $-\lambda$ vertices of $S$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Example 3.2
  • Corollary 3.3
  • Remark 1
  • ...and 11 more