Adjacency and Tensor Representation in General Hypergraphs.Part 2: Multisets, Hb-graphs and Related e-adjacency Tensors

HyperBagGraphs (hb-graphs as short) extend hypergraphs by allowing the hyperedges to be multisets. Multisets are composed of elements that have a multiplicity. When this multiplicity has positive integer values, it corresponds to non ordered lists of potentially duplicated elements. We define hb-graphs as family of multisets over a vertex set; natural hb-graphs correspond to hb-graphs that have multiplicity function with positive integer values. Extending the definition of e-adjacency to natural hb-graphs, we define different way of building an e-adjacency tensor, that we compare before having a final choice of the tensor. This hb-graph e-adjacency tensor is used with hypergraphs.

Authors

• 9 publications
• 8 publications
• 20 publications

Adjacency between two vertices in graphs or hypergraphs is a pairwise re...
12/21/2017 ∙ by Xavier Ouvrard, et al. ∙ 0

In graphs, the concept of adjacency is clearly defined: it is a pairwise...
09/01/2018 ∙ by Xavier Ouvrard, et al. ∙ 0

• Adjacency Matrix and Co-occurrence Tensor of General Hypergraphs: Two Well Separated Notions

Adjacency and co-occurrence are two well separated notions: even if they...
12/21/2017 ∙ by Xavier Ouvrard, et al. ∙ 0

• Optimal labelling schemes for adjacency, comparability and reachability

We construct asymptotically optimal adjacency labelling schemes for ever...
12/03/2020 ∙ by Marthe Bonamy, et al. ∙ 0

• Three hypergraph eigenvector centralities

Eigenvector centrality is a standard network analysis tool for determini...
07/25/2018 ∙ by Austin R. Benson, et al. ∙ 0

• Energized simplicial complexes

For a simplicial complex with n sets, let W^-(x) be the set of sets in G...
08/19/2019 ∙ by Oliver Knill, et al. ∙ 0

• Fast Generation of Unlabelled Free Trees using Weight Sequences

In this paper, we introduce a new representation for ordered trees, the ...
10/14/2020 ∙ by Paul Brown, et al. ∙ 0

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1 Introduction

Hypergraphs were introduced in Berge and Minieka (1973). Hypergraphs are defined as a family of nonempty subsets - called hyperedges - of the set of vertices. Elements of a set are unique. Hence elements of a given hyperedge are also unique in a hypergraph.

Multisets extend sets by allowing duplication of elements. As mentioned in Singh et al. (2007), N.G. de Bruijn proposed to Knuth the terminology multiset in replacement of a variety of existing terms, such as bag or weighted set. Multisets are used in database modelling: in Albert (1991) relational algebra extension basements were introduced to manipulate bags - see also Klug (1982) - by studying bags algebraic properties. Queries for such bags have been largely studied in a series of articles - see references in Grumbach et al. (1996): bags are intensively used in database queries as duplicate search is a costly operation. In Hernich and Kolaitis (2017) information integration under bag semantics is studied as well as the tractability of some algorithmic problems over bag semantics: they showed that the GLAV (Global-And-Local-As-View) mapping of two databases problem becomes untractable over such semantic. In Radoaca (2015a) and Radoaca (2015b), the author extensively study multisets and propose to represent them by two kind of Venn diagrams. Multisets are also used in P-computing in the form of labelled multiset, called membrane - see Păun (2006) for more details.

Taking advantage of this duplication allowance, we construct in this article an extension of hypergraphs called hyper-bag-graphs (shortcut as hb-graphs). There are two main reasons to get such an extension. The first reason is that multisets are extensively used in databases as they allow presence of duplicates Lamperti et al. (2000) - removing duplicates (and thus obtaining sets and hypergraphs) being an expensive operation. The second reason is that natural hb-graphs - hb-graphs based on multisets with non-negative integer multiplicity values - allow results on the hb-graph adjacency tensor: hypergraph being particular case of hb-graph, other hypergraph -adjacency tensors than the ones proposed in Banerjee et al. (2017); Ouvrard et al. (2017) can be built by giving meaningful interpretation to the steps taken during its construction via hb-graph.

Section 2 gives the mathematical background, including main definitions on hypergraphs and multisets. Section 3 gives mathematical construction of the Hyper-Bag-Graphs (or hb-graphs). Section 4 gives algebraic description of hb-graphs and consequences for the adjacency tensor of hypergraphs. Section 5 gives results on the constructed tensors. Section 6 evaluates the constructed tensors and proceed to a final choice on the hypergraph -adjacency tensor. Section 7 gives future work.

2 Mathematical background

2.1 Hypergraphs

As mentioned in Ouvrard et al. (2017), hypergraphs fit collaboration networks modelling - Newman (2001a, b) -, co-author networks - Grossman and Ion (1995), Taramasco et al. (2010) -, chemical reactions - Temkin et al. (1996) - , genome - Chauve et al. (2013) -, VLSI design - Karypis et al. (1999) - and other applications. More generally hypergraphs fit perfectly to keep entities grouping information. Hypergraphs succeed in capturing -adic relationships. In Berge and Minieka (1973), Stell (2012) and Bretto (2013) hypergraphs are defined in different ways. In this article, the definition of Bretto (2013) - as it doesn’t impose the union of the hyperedges to cover the vertex set - is used:

Definition 2.1.

An (undirected) hypergraph on a finite set of vertices (or vertices) is defined as a family of hyperedges where each hyperedge is a non-empty subset of .

A weighted hypergraph is a triple: where is a hypergraph and a mapping where each hyperedge is associated to a real number .

The -section of a hypergraph is the graph such that:

 ∀u∈V,∀v∈V:(u,v)∈E′⇔∃e∈E:u∈e∧v∈e

Let . A hypergraph is -uniform if all its hyperedges have the same cardinality .

A directed hypergraph is a hypergraph where each hyperedge accepts a partition in two non-empty subsets, called the source - written - and the target - written - with

Definition 2.2.

Let be a hypergraph.

The degree of a vertex is the number of hyperedges it belongs to. For a vertex , it is written or . It holds:

In this article only undirected hypergraphs will be considered. Hyperedges link one or more vertices together. Broadly speaking, the role of the hyperedges in hypergraphs is playing the role of edges in graphs.

2.2 Multisets

2.2.1 Generalities

Basic on multisets are given in this section, based mainly on Singh et al. (2007).

Definition 2.3.

Let be a set of distinct objects. Let

Let be an application from to .

Then is called a multiset - or mset or bag - on .

is called the ground or the universe of the multiset , is called the multiplicity function of the multiset .

is called the support - or root or carrier - of .

The elements of the support of a mset are called its generators.

A multiset where is called a natural multiset.

We write the set of all multisets of universe .

Some extensions of multisets exist where the multiplicity function can have its range in - called hybrid set in Loeb (1992). Some other extensions exist like fuzzy multisets Syropoulos (2000).

Several notations of msets exist. One common notation which we will use, if is the ground of a mset is to write:

where .

An other notation is:

 {x1,…,xn}m1,...,mn

or even:

 m1{x1}+…+mn{xn}.

If is a natural multiset an other notation is:

 ⎧⎪ ⎪⎨⎪ ⎪⎩⎧⎪ ⎪⎨⎪ ⎪⎩x1,…,x1m1times,…,xn,…,xnmntimes⎫⎪ ⎪⎬⎪ ⎪⎭⎫⎪ ⎪⎬⎪ ⎪⎭

which is similar to have an unordered list.

Remark 2.1.
1. Two msets can have same support and same support objects multiplicities but can differ by their universe.
Also to be equal two msets must have same universe, same support and same multiplicity function.

2. The multiplicity function corresponds to a weight that is associated to objects of the universe.

3. Multiplicity in natural multisets can also be interpreted as a duplication of support elements. In this case, a mset can be viewed as a non ordered list with repetition. In a natural multiset the copies of a generator of the support in instances are called elements of the multiset.

4. Some definitions of multisets also consider which could lead to interesting applications. We don’t develop such case here.

Definition 2.4.

Let be a mset.

The m-cardinality of written is defined as:

 #mAm=∑x∈Am(x).

The cardinality of - written is defined as:

 #Am=|A⋆m|.
Remark 2.2.

In general multisets, m-cardinality and cardinality are two separated notions as for instance: , and, have all same cardinality with different m-cardinalities for C compared to A and B.

In natural multisets, m-cardinality and cardinality are equal if and only if the multiplicity of each element in the support is 1, ie if the natural multiset is a set. It doesn’t generalize to general multisets - see A and B of the former example.

Definition 2.5.

Two msets and are said to be cognate if they have same support.

They are not necessarily equal: for instance, and are cognate but different.

Definition 2.6.

Let and be two msets on the same universe .

If is called the empty mset and written .

is said to be included in - written - if for all : . In this case, is called a submset of .

The union of and is the mset of universe and of multiplicity function such that for all :

 mC(x)=max(mA(x),mB(x)).

The intersection of and is the mset of universe and of multiplicity function such that for all :

 mD(x)=min(mA(x),mB(x)).

The sum of and is the mset of universe and of multiplicity function such that for all :

 mE(x)=mA(x)+mB(x).
Proposition 2.1.

, and are commutative and associative laws on msets of same universe. They have the empty mset of same universe as identity law.

is distributive for and .

and are distributive one for the other.

and are idempotent.

Definition 2.7.

Let be a mset.

The power set of , written , is the multiset of all submsets of .

2.2.2 Copy-set of a multiset

Let consider a multiset: where the range of the multiplicity function is a subset of . Equivalent definition - see Syropoulos (2000) - is to give a couple where is the set of all instances (including copies) of with an equivalency relation where:

 ∀x∈A0,∀x′∈A0:xρx′⇔∃!c∈A:xρc∧x′ρc.
Definition 2.8.

Two elements of such that: are said copies one of the other. The unique is called the original element. and are said copies of .

Also is isomorphic to and:

 ∀¯¯¯x∈A0/ρ,∃!c∈A:∣∣{x:x∈¯¯¯x}∣∣=m(c)∧∀x∈¯¯¯x:xρc.
Definition 2.9.

The set is called a copy-set of the multiset .

Remark 2.3.

A copy-set for a given multiset is not unique. Sets of equivalency classes of two couples and of a given multiset are isomorphic.

2.2.3 Algebraic representation of a multiset

We suppose given a natural multiset of universe and multiplicity function . It yields:

 Am={αm(αij)ij:αij∈A⋆m}.
Vector representation:

A multiset can be conveniently represented by a vector of length the cardinality of the universe and where the coefficients of the vector represent the multiplicity of the corresponding element.

Definition 2.10.

The vector representation of the multiset is the vector

This representation requires space and has null elements.

The sum of the elements of is

This representation will be useful later when considering family of multisets in order to build the incident matrix.

Hypermatrix representation:

An alternative representation is built by using a symmetric hypermatrix. This approach is needed to reach our goal of constructing an -adjacency tensor for general hypergraphs.

Definition 2.11.

The unnormalized hypermatrix representation of the multiset is the symmetrix hypermatrix of order and dimension such that if . The other elements are null.

Hence the number of non-zero elements in is out of the elements of the representation.

The sum of the elements of is then:

To achieve a normalisation, we enforce the sum of the elements of the hypermatrix to be the m-rank of the multiset it encodes. It yields:

Definition 2.12.

The normalized hypermatrix representation of the multiset is the symmetrix hypermatrix of order and dimension such that if . The other elements are null.

3 Hb-graphs

Hyper-bag-graphs - hb-graphs for short - are introduced in this section. Hb-graphs extend hypergraphs by allowing hyperedges to be msets. The goal of this section is to revisit some of the definitions and results found in Bretto (2013) for hypergraphs and extend them to hb-graphs.

3.1 Generalities

3.1.1 First definitions

Definition 3.1.

Let be a nonempty finite set.

A hyper-bag-graph - or hb-graph - is a family of msets with universe and support a subset of . The msets are called the hb-edges and the elements of the vertices.

We write the family of hb-edges and such a hb-graph.

We consider for the remainder of the article a hb-graph , with and the family of its hb-edges.

Each hb-edge is of universe and has a multiplicity function associated to it: where . When the context make it clear the notation is used for and for .

Definition 3.2.

A hb-graph is said with no repeated hb-edges if:

Definition 3.3.

A hb-graph where each hb-edge is a natural mset is called a natural hb-graph.

Remark 3.1.

For a general hb-graph each hb-edge has to be seen as a weighted system of vertices, where the weights of each vertex are hb-edge dependent.

In a natural hb-graph the multiplicity function can be viewed as a duplication of the vertices.

Definition 3.4.

The order of a hb-graph - written - is:

Its size is the cardinality of

Definition 3.5.

The empty hb-graph is the hb-graph with an empty set of vertices.

The trivial hb-graph is the hb-graph with a non empty set of vertices and an empty family of hb-edges.

If : then the hb-graph is said with no isolated vertices. Otherwise, the elements of are called the isolated vertices. They correspond to elements of hyperedges which have zero-multiplicity for all hb-edges.

Remark 3.2.

A hypergraph is a natural hb-graph where the vertices of the hb-edges have multiplicity one for any vertex of their support and zero otherwise.

3.1.2 Support hypergraph

Definition 3.6.

The support hypergraph of a hb-graph is the hypergraph whose vertices are the ones of the hb-graph and whose hyperedges are the support of the hb-edges in a one-to-one way. We write it , where .

Remark 3.3.

Given a hypergraph, an infinite set of hb-graphs can be generated that all have this hypergraph as support. To each of these hb-graphs corresponds a hb-edge family: to each support of these hb-edges corresponds at least a hyperedge in the hypergraph and reciprocally to each hyperedge corresponds at least a hb-edge in each hb-graph of the infinite set.

To have unicity, the considered hypergraph and hb-graphs should be respectively with no repeated hyperedge or with no repeated hb-edge.

3.1.3 m-uniform hb-graphs

Definition 3.7.

The m-range of a hb-graph - written - is by definition:

 rm(H)=maxe∈E#me.

The range of a hb-graph - written - is the range of its support hypergraph

The m-co-range of a hb-graph - written - is by definition:

 crm(H)=mine∈E#me.

The co-range of a hb-graph - written - is the range of its support hypergraph

Definition 3.8.

A hb-graph is said -m-uniform if all its hb-edges have same -cardinality .

A hb-graph is said -uniform if its support hypergraph is -uniform.

Proposition 3.1.

A hb-graph is -m-uniform if and only if:

 rm(H)=crm(H)=k.

Immediate.

3.1.4 HB-star and m-degree

Definition 3.9.

The HB-star of a vertex is the multiset - written - defined as:

 H(x)={eme(x):e∈E∧x∈e∗}.
Remark 3.4.

The support of the HB-star of a vertex of a hb-graph is exactly the star of this vertex in the support hypergraph .

Definition 3.10.

The m-degree of a vertex of a hb-graph - written - is defined as:

 degm(x)=#mH(x).

The maximal m-degree of a hb-graph is written .

The degree of a vertex of a hb-graph - written - corresponds to the degree of this vertex in the support hypergraph

The maximal degree of a hb-graph is written and corresponds to the maximal degree of the support hypergraph

Definition 3.11.

A hb-graph having all of its hb-edges of same m-degree is said m-regular or -m-regular.

A hb-graph is said regular if its support hypergraph is regular.

3.1.5 Dual of a hb-graph

Definition 3.12.

Considering a hb-graph , its dual is the hb-graph with a set of vertices which is in bijection with the set of hb-edges of :

 ∀~xi∈~V,∃!ei∈E:~xi=f(ei).

And the set of hb-edges is in bijection - where - with the set of vertices of .

Switching from the hb-graph to its dual:

Vertices
Edges
Multiplicity with with
-m-uniform -m-regular
-m-regular -m-uniform

3.2 Additional concepts for natural hb-graphs

3.2.1 Numbered copy hypergraph of a natural hb-graph

In natural hb-graphs the hb-edge multiplicity functions have their range in the natural number set. The vertices in a hb-edge with multiplicities strictly greater than 1 can be seen as copies of the original vertex.

Deepening this approach copies have to be understood as “numbered” copies. Let and be two hb-edges. Let be a vertex of multiplicity in and in . will hold copies: the ones “numbered” from 1 to . The remaining copies will be held either in xor depending which set has the highest multiplicity of .

More generally, we define the numbered-copy set of a multiset:

Definition 3.13.

Let .

The numbered copy-set of is the copy-set where: is a shortcut to indicate the numbered copies of the original element : to and is designated as the copy number of the element .

Definition 3.14.

Let be a natural hb-graph.

Let be the vertices of the hb-graph. Let be the hb-edges of the hb-graph and for , the multiplicity function of .

The maximum multiplicity function of is the function defined for all by:

 m(v)=maxe∈Eme(v).
Definition 3.15.

Let be a natural hb-graph where is the vertex set and is the hb-edge family of the hb-graph.

Let be the maximum multiplicity function.

Let consider the numbered-copy-set of the multiset :

Then each hb-edge is associated to a copy-set / equivalency relation which elements are in with copy number as small as possible for each vertex in .

Then where is a hypergraph called the numbered-copy-hypergraph of .

Proposition 3.2.

A numbered-copy-hypergraph is unique for a given hb-graph.

Proof.

It is immediate by the way the numbered-copy-hypergraph is built from the hb-graph.

Allowing the duplicates to be numbered prevent ambiguities; nonetheless it has to be seen as a conceptual approach as duplicates are entities that are not discernible.

3.2.2 Paths, distance and connected components

Defining a path in a hb-graph is not straightforward as vertices are duplicated in a hb-graph. The duplicate of a vertex strictly inside a path must be at the intersection of two consecutive hb-edges.

Definition 3.16.

A strict m-path in a hb-graph from a vertex to a vertex is a vertex / hb-edge alternation with hb-edges to and vertices to such that , , and and that for all , .

A large m-path from a vertex to a vertex is a vertex / hb-edge alternation with hb-edges to and vertices to such that , , and and that for all , .

is called in both cases the length of the m-path from to .

Vertices from to are called interior vertices of the m-path.

and are called extremities of the m-path.

If the extremities are different copies of the same object, then the m-path is said to be an almost cycle.

If the extremities designate exactly the same copy of one object, the m-path is said to be a cycle.

Remark 3.5.
1. For a strict m-path, there are:

possibilities of choosing the interior vertices along a given m-path and:

possible strict m-paths in between the extremities.

2. For a large m-path, there are:

possibilities of choosing the interior vertices along a given m-path and:

possible large m-paths in between the extremities.

3. As large m-paths between two extremities by a given sequence of interior vertices and hb-edges include strict m-paths, we often refer as m-paths for large m-paths.

4. If an m-path exists from to then an m-path also exists from to .

Definition 3.17.

An m-path in a hb-graph corresponds to a unique path in the hb-graph support hypergraph called the support path.

Proposition 3.3.

Every m-path traversing same hyperedges and having similar copy vertices as intermediate and extremity vertices share the same support path.

The notion of distance is similar to the one defined for hypergraphs.

Definition 3.18.

Let and be two vertices of a hb-graph. The distance from to is the minimal length of an m-path from to if such an m-path exists. If no m-path exist, and are said disconnected and .

Definition 3.19.

A hb-graph is said connected if its support hypergraph is connected, disconnected otherwise.

Definition 3.20.

A connected component of a hb-graph is a maximal set of vertices such that every pair of vertices of the component has an m-path in between them.

Remark 3.6.

A connected component of a hb-graph is a connected component of one of its copy hypergraph.

Definition 3.21.

The diameter of a hb-graph - written - is defined as:

 diam(H)=maxx,y∈Vd(x,y).

Definition 3.22.

Let be a positive integer.

Let consider vertices not necessarily distinct belonging to .

Let write the mset consisting of these vertices with multiplicity function .

The vertices are said -adjacent in if it exists such that .

Considering a hb-graph of m-range , the hb-graph can’t handle more than -adjacency in it. This maximal -adjacency is called the -adjacency of .

Definition 3.23.

Let consider a hb-edge in .

Vertices in the support of are said -adjacent.

Vertices in the hb-edge with nonzero multiplicity are said -adjacent.

Remark 3.7.
• -adjacency doesn’t support redundancy of vertices.

• -adjacency allows the redundancy of vertices.

• The only case of equality is where the hb-edge has all its nodes of multiplicity 1 at the most.

Definition 3.24.

Two hb-edges are said incident if their support intersection is not empty.

3.2.4 Sum of two hb-graphs

Let and be two hb-graphs.

The sum of two hb-graphs and is the hb-graph written defined as the hb-graph that has:

• as vertex set and where the hb-edges are obtained from the hb-edges of and with same multiplicity for vertices of (respectively ) but such that for each hyperedge in (respectively ) the universe is extended to and the multiplicity function is extended such that (respectively )

• as hb-edge family, ie the family constituted of the elements of and of the elements of .

 H1+H2=(V1∪V2,E1+E2)

This sum is said direct if doesn’t contain any new pair of repeated hb-edge than the ones already existing in and those already existing in . In this case the sum is written .

3.3 An example

Example 3.1.

Considering , with and with: , , , .

It holds:

 e1 e2 e3 e4 dm(vi) max{mej(vi)} v1 2 0 0 0 2 2 v2 0 3 0 0 3 3 v3 0 1 1 0 2 1 v4 2 0 0 0 2 1 v5 1 0 2 0 3 2 v6 0 0 0 1 1 1 v7 0 0 0 0 0 0 #mej 5 4 3 1

Therefore the order of is and its size is .

is an isolated vertex.

and are incident as well as and . is not incident to any hb-edge.

, and are -adjacent as they hold in .

, and are -adjacent as they hold in .

The dual of is the hb-graph: with:

• with for

• with:

has duplicated hb-edges and one empty hb-edge.

4 Algebraic representation of a hb-graph

4.1 Incidence matrix of a hb-graph

A mset is well defined by giving itself its universe, its support and its function of multiplicity. We have seen that a mset can be represented by a vector called the vector representation of the m-set.

Hb-edges of a given hb-graph have all the same universe.

Definition 4.1.

Let and be two positive integers.

Let be a non-empty hb-graph, with vertex set and .

The matrix is called the incidence matrix of the hb-graph .

This incidence matrix is intensively used in Ouvrard et al. (2018b) for diffusion by exchanges in hb-graphs.

4.2 e-adjacency tensor of a natural hb-graph

To build the -adjacency tensor of a natural hb-graph without repeated hb-edge - with vertex set and hb-edge set - we use a similar approach that was used in Ouvrard et al. (2017) using the strong link between cubical symmetric tensors and homogeneous polynomials.

Definition 4.2.

An elementary hb-graph is a hb-graph that has only one non repeated hb-edge in its hb-edge family.

Claim 4.1.

Let be a hb-graph with no repeated hb-edge.

Then:

 H=⨁e∈EHe

where is the elementary hb-graph associated to the hb-edge .

Proof.

Let and . As is with no repeated hb-edge, doesn’t contain new pairs of repeated elements. Thus is a direct sum.

A straightforward iteration over elements of leads trivially to the result.

We need first to define hypermatrices for the -adjacency of an elementary hb-graph and of a m-uniform hb-graph.

4.2.1 Normalised ¯¯¯k-adjacency tensor of an elementary hb-graph

We consider an elementary hb-graph where and is a multiset of universe and multiplicity function . The support of is by considering, without loss of generality: .

is the multiset: where .

The normalised hypermatrix representation of , written , describes uniquely the m-set . Thus the elementary hb-graph is also uniquely described by as is the unique hb-edge. is of rank and dimension .

Hence, the definition:

Definition 4.3.

Let be an elementary hb-graph with and the multiset of m-rank , universe and multiplicity function

The normalised -adjacency hypermatrix of an elementary hb-graph is the normalised representation of the multiset : it is the symmetric hypermatrix of rank and dimension where the only nonzero elements are:

where

In a elementary hb-graph the -adjacency corresponds to -adjacency. This hypermatrix encodes the -adjacency of the elementary hb-graph; as the -adjacency corresponds to -adjacency in such a hb-graph is encodes also the -adjacency of the elementary hb-graph.

4.2.2 hb-graph polynomial

Homogeneous polynomial associated to a hypermatrix:

With a similar approach than in Ouvrard et al. (2017) where full details are given, let write the canonical basis of .

is a basis of , where is the Segre outerproduct.

A tensor is associated to an hypermatrix by writting as:

Considering variables attached to the vertices and , the multilinear matrix product is a polynomial 111As a reminder::

of degree .

Elementary hb-graph polynomial:

Considering a hb-graph with and the multiset of m-rank , universe and multiplicity function .

Using the normalised -adjacency hypermatrix , which is symmetric, we can write the reduced version of its attached homogeneous polynomial :

 Pe(z0)=r!mj1!…mjk!qjmj11…jmjkkzmj1j1…zmjkjk.
Hb-graph polynomial:

Considering a hb-graph with no-repeated hb-edge, with and

This hb-graph can be summarized by a polynomial of degree :

 P(z0) = =

where is a technical coefficient. is called the hb-graph polynomial. The choice of is made in order to retrieve the m-degree of the vertices from the -adjacency tensor.

4.2.3 ¯¯¯k-adjacency hypermatrix of a m-uniform natural hb-graph

We now extend to m-uniform hb-graph the -adjacency hypermatrix obtained in the case of an elementary hb-graph.

In the case of a -m-uniform natural hb-graph with no repeated hb-edge, each hb-edge has the same -cardinality . Hence the -adjacency of a -m-uniform hb-graph corresponds to -adjacency where is the m-rank of the hb-graph. The -adjacency tensor of the hb-graph has rank and dimension . The elements of the -adjacency hypermatrix are:

 ai1…ir

with .

The associated hb-graph polynomial is homogeneous of degree

We obtain the definition of the -adjacency tensor of a -m-uniform hb-graph by summing the -adjacency tensor attached to each hyperedge with a coefficient equals to 1 for each hyperedge.

Definition 4.4.

Let be a hb-graph. .

The -adjacency hypermatrix of a -m-uniform hb-graph is the hypermatrix defined by:

where is the -adjacency hypermatrix of the elementary hb-graph associated to the hb-edge .

The only non-zero elements of are the elements of indices obtained by permutation of the multiset and are all equals to .

Remark 4.1.

When a -m-uniform hb-graph has 1 as vertex multiplicity for any vertices in each hb-edge support of all hb-edges, then this hb-graph is a -uniform hypergraph: in this case, we retrieve the result of the degree-normalized tensor defined in Cooper and Dutle (2012).

Claim 4.2.

The -degree of a vertex in a -m-uniform hb-graph of -adjacency hypermatrix is:

Proof.

has non-zero terms only for corresponding hb-edges that have in it. For such a hb-edge containing , it is described by . It means that the multiset corresponds exactly to the multiset For each such that there is possible permutation of the indices to and .

Also: .

4.2.4 Elementary operations on hb-graphs

In Ouvrard et al. (2017), we describe two elementary operations that are used in the hypergraph uniformisation process. We describe here two similar operations and some additional operations for hb-graphs.

Operation 4.1.
Let be a hb-graph. Let be a constant weighted function on hb-edges with constant value . The weighted hb-graph is called the canonical weighted hb-graph of The application is called the canonical weighting operation.

Operation 4.2.
Let be a canonical weighted hb-graph. Let . Let be a constant weighted function on hb-edges with constant value . The weighted hb-graph is called the -dilatated hb-graph of The application is called the c-dilatation operation.

Operation 4.3.
Let be a weighted hb-graph. Let be a new vertex. The y-complemented hbgraph of is the hbgraph where , - with the map such that for all , and is the multiset , with - and, the weight function is is such that : The application is called the y-complemented operation.

Operation 4.4.
Let be a weighted hb-graph. Let be a new vertex. Let The -vertex-increased hbgraph of is the hbgraph where , - with the map such that for all , and is the multiset , with - and, the weight function is is such that : The application is called the -vertex-increasing operation.

Operation 4.5.
The merged hb-graph of a family of weighted hb-graphs with is the weighted hb-graph with vertex set , with hb-edge family 222 is the family obtained with all elements of each family - with the map such that for all , and is the multiset , with - and, such that , The application is called the merging operation.

Operation 4.6.
Decomposing a hb-graph into a family of hb-graphs , where such that is called the decomposition operation .

Remark 4.2.

The direct sum of two hb-graphs appears as a merging operation of the two hb-graphs.

Definition 4.5.

Let and be two hb-graphs.

Let .

is said preserving