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

by   Xavier Ouvrard, et al.

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.


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1 Adjacency in hypergraphs

A hypergraph is a hyperedge family 111 is and is . is the permutation set on . over the vertex set [1]. A hypergraph with no repeated hyperedge is a hypergraph where the hyperedges are distinct pairwise.

We write the range of the hypergraph.

Hyperedge multi-adicity calls for additional adjacency concepts.

vertices are said -adjacent if it exists an hyperedge that contains them. Vertices of a given hyperedge are said e-adjacent. The -adjacency of an hypergraph is the maximal value of such that it exists vertices of the hypergraph that are -adjacent.

Hypermatrices - abusively designated as tensors [2] - are used to store the adjacency multi-adic relationships. In -uniform hypergraphs, where all hyperedges have the same cardinality , -adjacency and e-adjacency are equivalent; we use here the degree normalized -adjacency hypermatrix [3].

For general hypergraphs with no-repeated hyperedge, a first e-adjacency hypermatrix is defined in [4]. The value and the number of elements that are required to store this hypermatrix vary depending on the hyperedge cardinality; due to index repetition, tensor elements can not be interpreted directly in term of a hypergraph uniformisation process (HUP). To address this issue, we propose a new e-adjacency tensor222Details and proofs can be found in [5]..

2 A new e-adjacency tensor for general hypergraphs

We give here only the main steps.333Exponents into parenthesis refer to the order of the corresponding tensor; indices into parenthesis refer to a sequence of objects.

2.1 Decomposition in layers

The family where constitutes a partition of . is decomposable uniquely into a -uniform hypergraph direct sum of increasing . The - - are called the layers of . Any of these is representable by a degree-normalised -adjacency hypermatrix .

Symmetric cubical hypermatrices are bijectively mapped to homogeneous polynomials [6] through the hypermatrix multilinear matrix multiplication [7].

We build a family of homogenous polynomials that are one-to-one mapped to the layers of the hypergraph. Considering 444We write the variable list and the variable list . - for all : represents - and , contains only one element: As is symmetric: with

2.2 Uniformisation and homogeneisation process

The hypergraph uniformisation process involves two elementary operations on weighted hypergraphs.

Operation 1: Let be a weighted hypergraph. Let . The -vertex-augmented hypergraph of is the weighted hypergraph where , - with the map such that: - and, such that :

Operation 2: The merged hypergraph of two weighted hypergraphs and is the weighted hypergraph with vertex set , with hyperedge family - constituted of all elements of and all elements of - such that , and ,

The hypergraph uniformisation process starts by mapping each to a weighted hypergraph with: with and . are dilatation coefficients introduced to guarantee that the generalized hand-shake lemma holds in the e-adjacency tensor. A set of pairwise distinct vertices is generated and such that no vertex of is in .

The HUP iterates over a two-phase step: the inflation phase (IP) and the merging phase (MP). At step the input is the -uniform weigthed hypergraph obtained from the previous iteration; at step 1, In the IP, is transformed into the -uniform -vertex-augmented hypergraph of .

The MP elaborates the merged hypergraph from and

At the end of each step is increased until it reaches : the last obtained is called the -layered uniform hypergraph of .

captures exactly the e-adjacency of .

In the polynomial homogeneisation process, the family of homogeneous polynomials of degree is obtained iteratively from the family : for all , maps one to one to .

We set We generate new pairwise distinct variables , .

At step , we suppose that: with the convention that: if Then for :

and for :

Even if the step above is performed: the degree of will increase by 1.

2.3 Construction of the e-adjacency tensor

From we build a symmetric tensor. is an homogeneous polynomial with variables of order . With for , we have: where:

  • for : and for :

  • for all , for , for all 555With the convention if : and, for all :

  • otherwise is null.

Also can be linked to a symmetric hypercubic tensor of order and dimension written whose elements are .

The coefficients , are chosen so that the number of edges calculated by the generalized handshake lemma is valid.

We choose: as:

Hence, combining above with the fact that when and 0 otherwise: for nonzero elements of .

The hypermatrix is called the layered e-adjacency tensor of the hypergraph . We write it later

3 Further comments and results

The HUP adds vertices in the IPs; they give indication on the original cardinality of the hyperedge they are added to as well as the level of -adjacency possible in this hyperedge. The resulting tensor is symmetric and is bijectively associated to the original hypergraph, containing its overall structure.

We consider in the following propositions a hypergraph with no repeated hyperedge with layered e-adjacency tensor

It holds:

where: and

Moreover: :


Using the definition of eigenvalue of

[2], we state:

The e-adjacency tensor has its eigenvalues such that:


where and

Let be a -regular666A hypergraph is said -regular if all vertices have same degree . -uniform hypergraph with no repeated hyperedge. Then this maximum is reached.

4 Conclusion

Properly defining the concept of adjacency in a hypergraph is important to build a proper e-adjacency tensor that preverves the information on the structure of the hypergraph. The resulting tensor allows to reconstruct with no ambiguity the original hypergraph. First results on spectral analysis show that additional vertices inflate the spectral radius bound. The HUP is a strong basis for further proposals: to allow repetition of vertices, we introduce hb-graphs, family of multisets and, propose two other e-adjacency tensors [8].