On some building blocks of hypergraphs

TL;DR

This paper investigates the spectral properties of hypergraphs through symmetry-based equivalence relations, introducing new structural components called building blocks that influence hypergraph spectra and related properties.

Contribution

It introduces the concept of $rak{R}_s$-compatible operators and identifies hypergraph building blocks like units and twin units, linking these to spectral and structural features.

Findings

Eigenvalues can be derived from equivalence classes or quotient operators.

Presence of building blocks affects the spectrum and eigenspaces.

Spectral footprints reveal hypergraph symmetries and structures.

Abstract

In this study, we explore the interrelation between hypergraph symmetries represented by equivalence relations on the vertex set and the spectra of operators associated with the hypergraph. We introduce the idea of equivalence relation compatible operators related to hypergraphs. Some eigenvalues and the corresponding eigenvectors can be computed directly from the equivalence classes of the equivalence relation. The other eigenvalues can be computed from a quotient operator obtained by identifying each equivalence class as an element. We provide an equivalence relation $\mathfrak{R}_s$ on the vertex set of a hypergraph such that the Adjacency, Laplacian, and signless Laplacian operators associated with that hypergraph become $\mathfrak{R}_s$-compatible. The $\mathfrak{R}_s$-equivalence classes are named as units. Using units, we find some more symmetric substructures of hypergraphs called twin units, regular sets, co-regular sets, and symmetric sets. We collectively classify them as building blocks of hypergraphs. We show that the presence of these building blocks leaves certain traces in the spectrum and the corresponding eigenspaces of the $\mathfrak{R}_s$-compatible operators associated with the hypergraph. We also show that, conversely, some specific footprints in the spectrum and the corresponding eigenvectors retrace the presence of some of these building blocks in the hypergraph. Besides the spectra of $\mathfrak{R}_s$-compatible operators, building blocks are also interrelated with hypergraph colouring, distances in hypergraphs, hypergraph automorphisms, and random walks on hypergraphs.