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

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