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

Published in Electronic Notes in Discrete Mathematics. 2018, vol. 70, p. 71-76
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.
Keywords HypergraphE-adjacency tensorUniformisationHomogeneisation
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Research groups Computer Vision and Multimedia Laboratory
Viper group
(ISO format)
OUVRARD, Xavier Eric, LE GOFF, Jean-Marie, MARCHAND-MAILLET, Stéphane. On Adjacency and e-Adjacency in General Hypergraphs: Towards a New e-Adjacency Tensor. In: Electronic Notes in Discrete Mathematics, 2018, vol. 70, p. 71-76. doi: 10.1016/j.endm.2018.11.012 https://archive-ouverte.unige.ch/unige:129063

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Deposited on : 2020-01-17

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