1 Adjacency in hypergraphs
A hypergraph is a hyperedge family ^{1}^{1}1 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 multiadicity calls for additional adjacency concepts.
vertices are said adjacent if it exists an hyperedge that contains them. Vertices of a given hyperedge are said eadjacent. 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 multiadic relationships. In uniform hypergraphs, where all hyperedges have the same cardinality , adjacency and eadjacency are equivalent; we use here the degree normalized adjacency hypermatrix [3].
For general hypergraphs with norepeated hyperedge, a first eadjacency 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 eadjacency tensor^{2}^{2}2Details and proofs can be found in [5]..
2 A new eadjacency tensor for general hypergraphs
We give here only the main steps.^{3}^{3}3Exponents 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 degreenormalised 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 onetoone mapped to the layers of the hypergraph. Considering ^{4}^{4}4We 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 vertexaugmented 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 handshake lemma holds in the eadjacency tensor. A set of pairwise distinct vertices is generated and such that no vertex of is in .
The HUP iterates over a twophase 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 vertexaugmented 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 eadjacency 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 eadjacency 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 ^{5}^{5}5With 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 eadjacency 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 eadjacency tensor
It holds:
where: and
Moreover: :
and:
Using the definition of eigenvalue of
[2], we state:The eadjacency tensor has its eigenvalues such that:
(1) 
where and
Let be a regular^{6}^{6}6A 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 eadjacency 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 hbgraphs, family of multisets and, propose two other eadjacency tensors [8].
References
 [1] A. Bretto, Hypergraph theory, An introduction. Mathematical Engineering. Cham: Springer.
 [2] L. Qi, Z. Luo, Tensor analysis: spectral theory and special tensors, Vol. 151, SIAM, 2017.
 [3] J. Cooper, A. Dutle, Spectra of uniform hypergraphs, Linear Algebra and its Applications 436 (9) (2012) 3268–3292.
 [4] A. Banerjee, A. Char, B. Mondal, Spectra of general hypergraphs, Linear Algebra and its Applications 518 (2017) 14–30.
 [5] X. Ouvrard, J.M. Le Goff, S. MarchandMaillet, Adjacency and tensor representation in general hypergraphs part 1: eadjacency tensor uniformisation using homogeneous polynomials, arXiv preprint arXiv:1712.08189.
 [6] P. Comon, Y. Qi, K. Usevich, A polynomial formulation for joint decomposition of symmetric tensors of different orders, in: International Conference on Latent Variable Analysis and Signal Separation, Springer, 2015, pp. 22–30.
 [7] L.H. Lim, Tensors and hypermatrices, Handbook of Linear Algebra, 2nd Ed., CRC Press, Boca Raton, FL (2013) 231–260.
 [8] X. Ouvrard, J.M. L. Goff, S. MarchandMaillet, Adjacency and tensor representation in general hypergraphs. part 2: Multisets, hbgraphs and related eadjacency tensors, arXiv preprint arXiv:1805.11952.
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