Adjacency Matrix and Co-occurrence Tensor of General Hypergraphs: Two Well Separated Notions

Adjacency and co-occurence are two well separated notions: even if they are the same for graphs, they start to be two different notions for uniform hypergraphs. After having done the difference between the two notions, this paper contributes in the definition of a co-occurence tensor reflecting the general hypergraph structure. It is a challenging issue that can have many applications if properly achieved, as it will allow the study of the spectra of such general hypergraph. In most of the applications only an hypermatrix associated to the tensor is needed. In this article, a novel way of building a symmetric co-occurence hypermatrix is proposed that captures also the cardinality of the hyperedges and allows full separation of the different layers of the hypergraph.