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All eigenvalues of the blowup of a graph

Ge Lin, Changjiang Bu

TL;DR

The paper addresses the problem of determining all eigenvalues of the $s$-blowup of a graph, a $2s$-uniform hypergraph obtained by vertex expansion. It introduces $2s$-weighted graphs and proves a bidirectional correspondence: every eigenvalue $\lambda$ of the blowup $G^{[s]}$ arises from some $2s$-weighted induced subgraph of $G$, and conversely, such eigenvalues yield eigenvalues of $G^{[s]}$. This yields a complete eigenvalue characterization and, in particular, shows $\rho(G^{[s]})=\rho(G)$; the paper also provides explicit eigenvalues for $K_{3}^{[2]}$ and discusses real $\mathrm{N}$-eigenvalues, highlighting how hypergraph spectra can be reduced to weighted subgraph spectra. The results supply a unifying, constructive method for spectral analysis of blowups and related hypergraphs with potential applications in spectral graph theory and hypergraph theory.

Abstract

The $s$-blowup of a graph ($s\geq2$) is the $2s$-uniform hypergraph obtained by replacing each vertex with a set of size $s$ and preserving the adjacency relation. In this paper, we define $2s$-weighted graphs and use them to give all eigenvalues of the $s$-blowup of a graph.

All eigenvalues of the blowup of a graph

TL;DR

The paper addresses the problem of determining all eigenvalues of the -blowup of a graph, a -uniform hypergraph obtained by vertex expansion. It introduces -weighted graphs and proves a bidirectional correspondence: every eigenvalue of the blowup arises from some -weighted induced subgraph of , and conversely, such eigenvalues yield eigenvalues of . This yields a complete eigenvalue characterization and, in particular, shows ; the paper also provides explicit eigenvalues for and discusses real -eigenvalues, highlighting how hypergraph spectra can be reduced to weighted subgraph spectra. The results supply a unifying, constructive method for spectral analysis of blowups and related hypergraphs with potential applications in spectral graph theory and hypergraph theory.

Abstract

The -blowup of a graph () is the -uniform hypergraph obtained by replacing each vertex with a set of size and preserving the adjacency relation. In this paper, we define -weighted graphs and use them to give all eigenvalues of the -blowup of a graph.
Paper Structure (3 sections, 1 theorem, 19 equations)

This paper contains 3 sections, 1 theorem, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Theorem 3.1

Let $G$ be a graph and let $s\geq2$. A complex number $\lambda$ is an eigenvalue of $G^{[s]}$ if and only if $\lambda$ is an eigenvalue of some $2s$-weighted induced subgraph of $G$.

Theorems & Definitions (3)

  • Definition 2.1
  • Theorem 3.1
  • proof