On Adjacency and e-Adjacency in General Hypergraphs: Towards a New e-Adjacency Tensor

TL;DR

This paper introduces a new e-adjacency tensor for hypergraphs, capturing complex multi-adic relationships through hypergraph uniformisation and polynomial homogenisation, enabling advanced spectral analysis.

Contribution

It proposes a novel e-adjacency tensor for hypergraphs, integrating hypergraph uniformisation and polynomial homogenisation to improve interpretability and structural analysis.

Findings

The tensor is symmetric and fully described by hyperedge count.

It captures hypergraph structure including maximum k-adjacency levels.

Spectral analysis results demonstrate its analytical potential.

Abstract

In graphs, the concept of adjacency is clearly defined: it is a pairwise relationship between vertices. Adjacency in hypergraphs has to integrate hyperedge multi-adicity: the concept of adjacency needs to be defined properly by introducing two new concepts: $k$-adjacency - $k$ vertices are in the same hyperedge - and e-adjacency - vertices of a given hyperedge are e-adjacent. In order to build a new e-adjacency tensor that is interpretable in terms of hypergraph uniformisation, we designed two processes: the first is a hypergraph uniformisation process (HUP) and the second is a polynomial homogeneisation process (PHP). The PHP allows the construction of the e-adjacency tensor while the HUP ensures that the PHP keeps interpretability. This tensor is symmetric and can be fully described by the number of hyperedges; its order is the range of the hypergraph, while extra dimensions allow to capture additional hypergraph structural information including the maximum level of $k$-adjacency of each hyperedge. Some results on spectral analysis are discussed.