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On Adjacency and e-adjacency in General Hypergraphs: Towards an e-adjacency Tensor

by   Xavier Ouvrard, et al.

Adjacency between two vertices in graphs or hypergraphs is a pairwise relationship. It is redefined in this article as 2-adjacency. In general hypergraphs, hyperedges hold for n-adic relationship. To keep the n-adic relationship the concepts of k-adjacency and e-adjacency are defined. In graphs 2-adjacency and e-adjacency concepts match, just as k-adjacency and e-adjacency do for k-uniform hypergraphs. For general hypergraphs these concepts are different. This paper also contributes in a uniformization process of a general hypergraph to allow the definition of an e-adjacency tensor, viewed as a hypermatrix, reflecting the general hypergraph structure. This symmetric e-adjacency hypermatrix allows to capture not only the degree of the vertices and the cardinality of the hyperedges but also makes a full separation of the different layers of a hypergraph.


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