All eigenvalues of the blowup of a graph

TL;DR

This paper provides a complete characterization of the eigenvalues of the s-blowup of a graph using the concept of 2s-weighted graphs, advancing spectral graph theory.

Contribution

It introduces 2s-weighted graphs and derives all eigenvalues of the s-blowup of a graph, offering a new spectral analysis method.

Findings

All eigenvalues of the s-blowup are explicitly determined.

The method applies to any s ≥ 2.

Provides a new tool for spectral analysis of hypergraph constructions.

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.