Symmetries of Hypergraphs and Some Invariant Subspaces of Matrices Associated with Hypergraphs

TL;DR

This paper explores how the symmetries of hypergraphs influence the eigenvalues and eigenvectors of associated matrices, revealing invariant subspaces that relate to structural symmetries and dynamical processes.

Contribution

It introduces a framework linking hypergraph symmetries to invariant subspaces of matrices, advancing understanding of structural effects on spectral properties.

Findings

Certain matrices reflect hypergraph symmetries in their eigenstructure

Invariant subspaces correspond to vertex clusters affected by symmetries

Symmetries influence dynamical behaviors like random walks on hypergraphs

Abstract

Here, the structural symmetries of a hypergraph are represented through equivalence relations on the vertex set of the hypergraph. A matrix associated with the hypergraph may not reflect a specific structural symmetry. In the context of a given symmetry within a hypergraph, we investigate a collection of matrices that encapsulate information about the symmetry. Our investigation reveals that certain structural symmetries in a hypergraph manifest observable effects on the eigenvalues and eigenvectors of designated matrices associated with the hypergraph. We identify specific matrices where the invariance is a consequence of symmetries present in the hypergraph. These invariant subspaces elucidate analogous behaviours observed in certain clusters of vertices during random walks and other dynamical processes on the hypergraph.