On Hb-graphs and their application to general hypergraph e-adjacency tensor

Abstract

Working on general hypergraphs requires to properly redefine the concept of adjacency in a way that it captures the information of the hyperedges independently of their size. Coming to represent this information in a tensor imposes to go through a uniformisation process of the hypergraph. Hypergraphs limit the way of achieving it as redundancy is not permitted. Hence, our introduction of hb-graphs, families of multisets on a common universe corresponding to the vertex set, that we present in details in this article, allowing us to have a construction of adequate adjacency tensor that is interpretable in term of $m$-uniformisation of a general hb-graph. As hypergraphs appear as particular hb-graphs, we deduce two new ($e$)-adjacency tensors for general hypergraphs. We conclude this article by giving some first results on hypergraph spectral analysis of these tensors and a comparison with the existing tensors for general hypergraphs, before making a final choice.