Adjacency and Tensor Representation in General Hypergraphs Part 1: e-adjacency Tensor Uniformisation Using Homogeneous Polynomials
Abstract
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 -adic relationship. To keep the -adic relationship the concepts of -adjacency and e-adjacency are defined. In graphs 2-adjacency and e-adjacency concepts match, just as -adjacency and e-adjacency do for -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.
Keywords
Cite
@article{arxiv.1712.08189,
title = {Adjacency and Tensor Representation in General Hypergraphs Part 1: e-adjacency Tensor Uniformisation Using Homogeneous Polynomials},
author = {Xavier Ouvrard and Jean-Marie Le Goff and Stéphane Marchand-Maillet},
journal= {arXiv preprint arXiv:1712.08189},
year = {2018}
}