Spectral graph theory is a well-known devolved area of research in mathematics where one of the interests is to investigate the relationship between the spectrum of different matrices representing graphs with their structure. We also propose various graph operations, for example, graph joining, etc., and find the eigenvalues of newly generated graphs by those operations. Here we explore the same for hypergraphs.
As graphs, hypergraphs also have a vertex set and an edge set, but, unlike a graph, an edge of a hypergraph is a nonempty subset of its vertex set. Hypergraphs can be represented by different hypermatrices or tensors[4, 16] as well by various matrices [22, 27]. Here we consider a matrix representation of hypergraphs suggested in  and find the spectrum of hypergraphs formed by joining operations. In graph theory, graph joining  plays an important role in constructing certain graphs. Here we define a general hypergraph joining operation and extend it in different scenarios, such as a weighted or un-weighted joining of a set of weighted or un-weighted hypergraphs, the building of a hypergraph of hypergraphs, etc. We reframe the concept of equitable partition defined on graphs [18, 24, 26], for hypergraphs to investigate the spectrum of different joins of hypergraphs. A hypergraph consists of only the edges with the same cardinality , is called -uniform hypergraph. An -uniform hypergraph (on vertices) having all possible edges is called a complete -uniform hypergraph and is denoted by . Furthermore is called the complement of which is also a complete -uniform hypergraph with vertices such that is an edge of if and only if it is not an edge of . A graph is known as complete multi()-partite if the vertex set is partitioned into more than one () parts and the edge set consists of only the all possible edges with one terminal vertex is in one partite and another terminal vertex is in a different partite. Complete multipartite graphs contain a class of integral graphs [20, 25, 21]. It is still challenging to find all the eigenvalues of a complete multipartite graph, although its characteristic polynomial is known . Now, a complete -uniform -partite hypergraph, , with the vertex set partitioned into parts say , is called complete -uniform weak(strong) -partite hypergraph if () and the edge set . Here, we are interested to compute the characteristic polynomial of a complete -uniform -partite hypergraph . The concept of corona of two graphs was introduced by R. Fruchut and F. Harary . Here we also defined the generalized corona of hypergraphs. Vertex corona of two graphs can provide infinitely many pairs of non-isomorphic co-spectral graphs [6, 5, 10, 12]. We derive the same for two hypergraphs. Edge corona, which is an interesting graph operation [11, 8], is defined for hypergraphs and is applied to find the eigenvalues of -loose paths and -loose cycles. Generalized -loose path is an -uniform hypergraph with the vertex set and edge set the . For , is known as loose path. Similarly, generalized -loose cycle is an -uniform hypergraph with the vertex set and the edge set . In spectral study of hypergraphs, using hypermatrix, it is not easy to find the eigenvalues of , hyper-cubes. In , the authors considered oriented hypergraphs and proved some results on the spectrum of adjacency matrices of . They also posed some open questions regarding the spectrum of . Here we find the spectrum of for . We also compute some eigenvalues of an -loose path and the remaining eigenvalues are shown as the zeros of a polynomial.
In this article, the Perron-Frobenius theorem for nonnegative irreducible matrices has been extensively used.
Lemma 1 (Perron-Frobenius Theorem, Chapter 8,  ).
Let be a non-negative irreducible matrix. Then
has a positive eigenvalue
with positive eigenvector.
is simple and for any other eigenvalue of ,
This positive eigenvalue known as Perron eigenvalue and others as non-Perron eigenvalue.
2 Adjacency Matrix for Hypergraphs
Let be a hypergraph with the vertex set the edge set and with an weight function defined by for all The adjacency matrix of is defined as
This definition is adopted from the definition of the adjacency matrix of an unweighted non-uniform hypergraph defined in . The valency of a vertex of is given by . Thus for an -uniform hypergraph we have . If we take , then where is the codegree of the vertices , i.e., the number of edges containing both the vertices and 111In 2002 Rodriguez defined the adjacency matrix and corresponding Laplacian matrix for uniform hypergraphs and which is the matrix .. If the vertices and belong to an edge we call, they are adjacent and denote as , otherwise, . For unweighted hypergraph we take for all and then the hypergraph is denoted by . For an unweighted hypergraph the valency of a vertex is the degree of , i.e., the number of edges containing the vertex . A hypergraph is (-)regular if all the degrees of vertices are equal( to ). A hypergraph is connected if for any two vertices there is an alternating sequence of distinct vertices and distinct edges , such that, for . For an -regular connected hyperghraph , with vertices is always an eigenpair of and by Lemma 1, is a simple eigenvalue of . Moreover, since is real symmetric matrix, thus we can have an orthogonal set of eigenvectors for .
2.1 Equitable Partition for Hypergraphs
Let be an -uniform hypergraph. We say a partition of is an equitable partition of if for any and for any ,
where is a constant depends only on and . Note that any two vertices belonging to the same cell of the partition have the same valency. For an equitable partition with -number of cells we define the quotient matrix as , for .
Let be an -uniform hypergraph. Then orbits of any subgroup of form an equitable partition.
Let be a subgroup of acts on the vertex set . We consider the orbits as . For any there exists such that . Now we have
This completes the proof. ∎
Let be an -uniform weighted hypergraph with vertices. For an equitable partition with cells we define the characteristic matrix of order as follows
Thus . The column space of is -invariant. It is easy to show that is also diagonalizable as and for each with the multiplicity , with the multiplicity atlest .
3 Weighted Joining of Hypergraphs
3.1 Weighted Joining of a Set of Uniform Hypergraphs
First we consider the joining of two hypergraphs. Let and be two -uniform hypergraphs. The join of and is the hypergraph with the vertex set , edge set , where and the weight function is defined by
where is a real non-negative constant.
Now, we define the same operation for a set of -unoform hypergraphs. Let , , be a set of -uniform hypergraphs, . Using the set , of hypergraphs we construct a new -uniform hypergraph where , , and defined by
where is a real non-negative constant. The resultant hypergraph is called the join of a set of -uniform hypergraphs ’s and it is denoted as .
Now we write the adjacency matrix of the join, . Let and . Let
For , , we take two fixed vertices such that , and define
Therefore the adjacency matirx of can be expressed as
3.1.1 Weighted Joining of Uniform Hypergraphs on a Backbone Hypergraph
Let be an -uniform hyperghraph. We call the hypergraph , as a backbone of if can be constructed by a set : , of -uniform hypergraphs, with the following operations :
Replace vertex of by , for .
For each edge , take and apply the operation defined above.
We call the hypergraphs ’s as participants on the backbone to form the hypergraph . Thus the adjacency matrix for can be written as
Let be a hypergraph with the vertex set and let be a collection of regular -uniform hypergraphs . Let be the -uniform hypergraph constructed by taking as backbone hypergraph and ’s as participants. Then for any non-Perron eigenvalue of with multiplicity , is an eigenvalue of with the multiplicity atleast .
Let be an eigenpair of , such that
is orthiogonal to the constant vector. We define by
Thus is an eigenvector of corresponding to the eigenvalue . Since thus the proof folows. ∎
When ’s are regular, the partition forms an equitable partition for . In particular if ’s are regular then the quotient matrix is as follows
The remaining eigenvalues of can be obtained from .
Take , and , . Consider as the backbone and ’s as the participants. Then we get the complete -uniform -partite hypergraph . By the above theorem we have is an eigenvalue of with the multiplicity atleast . The quotient matrix is given by
where for and .
Let be a set of -uniform hypergraphs where are -regular. Let . Then for any non-perron eigenvalue of with the multiplicity , is an eigenvalue of with multiplicity atleast .
Note that the remaining eigenvalues can be obtained from the quotient martrix defined as
Let be an undirected weighted graph with the adjacency matrix and ’s be -regular -uniform hypergraphs, for . Let be the -uniform hypergraph constructed by taking as the backbone graph and ’s as the participants. Then for any non-perron eigenvalue of with multiplicity , is an eigenvalue of with the multiplicity atleast .
If we consider as the backbone and as the participants to construct the hypergraph then is written as .
Take , , and . Then , which is the weak -uniform -partite complete hypergraph. Using the above corollary we get that for any ; is an eigenvalue of with the multiplicity atleast for . The remaining eigenvalues of are the eigenvalues of the quotient matrix , defined as
In the above two examples, if we take , we have as an eigenvalue of with the multiplicity atleast . Here, the quotient matrix formed by the equitable partition and which is given by
Note that if and only if for .
Using from , we have the characteristic polynomial of is
We may consider as the generalization of complete bipartite graphs. Now we have the following proposition
Characteristic polynomial of is where , and .
From the above proposition it is clear that the quotient matrix is non-singular. Hence the multiplicity of eigenvalue of is .
Note that . Thus the non-zero eigenvalues of are .
Note: Let be the -uniform -partite hypergraph , where . Then the quotient matrix for the equitable partition formed by the -parts of can be written as , where is given by
where , . Now using the Lemma 2, we have the characteristic polynomial of as follows
where . Thus the eigenvalues of are with the multiplicity for and with the multiplicity .
3.2 Weighted Joining of a Set of Non-Uniform Hypergraphs
Let be a set of hypergraphs with . Now for , we define the join as a hypergraph with the vertex set , the edge set , where and the weight function defined by
where are non-negative constants for each The adjacency matirx of the hypergraph is given by
When ’s are regular, forms an equitable partition for . If ’s are -regular then the quotient matrix is given by
3.2.1 Weighted Joining of Non-Uniform Hypergraphs on a Backbone Hypergraph
Let be a hypergraph. We consider the hypergraph , with the vertex set as a backbone of if there exist a set