Spectra of general hypergraphs

Introduction

Spectral graph theory has a long history behind its development. In spectral graph theory, we analyse the eigenvalues of a connectivity matrix which is uniquely defined on a graph. Many researchers have had a great interest to study the eigenvalues of different connectivity matrices, such as, adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, etc. Now, a recent trend has been developed to explore spectral hypergraph theory. Unlike in a graph, an edge of a hypergraph can be constructed with more than two vertices, i.e., the edge set of a hypergraph is the subset of the power set of the vertex set of that hypergraph . Now, one of the main challenges is to uniquely represent a hypergraph by a connectivity hypermatrix or by a tensor, and vice versa. It is not trivial for a non-uniform hypergraph, where the cardinalities of the edges are not the same. Recently, the study of the spectrum of uniform hypergraph becomes popular. In a ($m$-) uniform hypergraph, each edge contains the same, ($m$), number of vertices. Thus an $m$-uniform hypergraph of order $n$ can be easily represented by an $m$ order $n$ dimensional connectivity hypermatrix (or tensor). In , the results on the spectrum of adjacency matrix of a graph are extended for uniform hypergraphs by using characteristic polynomial. Spectral properties of adjacency uniform hypermatrix are deduced from matroids in . In 1993, Fan Chung defined Laplacian of a uniform hypergraph by considering various homological aspects of hypergraphs and studied the eigenvalues of the same . In , different spectral properties of Laplacian and signless Laplacian of a uniform hypergraph, defined by using tensor, have been studied. In 2015, Hu and Qi introduced the normalized Laplacian of a uniform hypergraph and analyzed its spectral properties . The important tool that has been used in spectral hypergraph theory is tensor. In 2005, Liqun Qi introduced the different eigenvalues of a real supersymmetric tensor . The various properties of the eigenvalues of a tensor have been studied in .

But, still the challenge remains to come up with a mathematical framework to construct a connectivity hypermatrix for a non-uniform hypergraph, such that, based on this connectivity hypermatrix the spectral graph theory for a general hypergraph can be developed. Here, we propose a unique representation of a general hypergraph (without any self loop or multiple edge) by connectivity hypermatrices, such as, adjacency hypermatrix, Laplacian hypermatrix, signless Laplacian hypermatrix, normalized Laplacian hypermatrix and analyze the different spectral properties of these matrices. These properties are very similar with the same for graphs and uniform hypergraphs. Studying the spectrum of a uniform hypergraphs could be considered as a special case of the spectral graph theory of general hypergraphs.

Preliminary

We call $(\lambda,x)$ a $Z$-eigenpair if both of them are real.

From the above definitions it is clear that, a constant multiplication of an eigenvector is also an eigenvector corresponding to an $H$-eigenvalue, but, this is not always true for $E$-eigenvalue and $Z$-eigenvalue. Now, we recall some results that are used in the next section.

The above theorem helps us to bound the eigenvalues of a tensor.

Let $A$ be an $m$ order and $n$ dimensional tensor and $D=diag(d_{1},\dots,d_{n})$ be a positive diagonal matrix. Define a new tensor

Then $A$ and $B$ have the same $H$-eigenvalues.

Spectral properties of general hypergraphs

A (general) hypergraph $G$ is a pair $G=(V,E)$ where $V$ is a set of elements called vertices, and $E$ is a set of non-empty subsets of $V$ called edges. Therefore, E is a subset of $\mathcal{P}(V)\setminus\{\emptyset\}$, where $\mathcal{P}(V)$ is the power set of $V$.

Let $G=(V,E)$, where $V=\{1,2,3,4,5\}$ and E=\big{\{}\{1\},\{2,3\},\{1,4,5\}\big{\}}. Here, $G$ is a hypergraph of $5$ vertices and $3$ edges.

Let $G=(V,E)$ be the hypergraph where $V=\{v_{1},v_{2},\dots,v_{n}\}$ and $E=\{e_{1},e_{2},\dots,e_{k}\}$. Let $m=max\{|e_{i}|:e_{i}\in E\}$ be the maximum cardinality of edges, $m.c.e(G)$, of $G$. Define the adjacency hypermatrix of $G$ as

For all edges $e=\{v_{l_{1}},v_{l_{2}},\dots,v_{l_{s}}\}\in E$ of cardinality $s\leq m$,

and $p_{1},p_{2},\dots,p_{m}$ chosen in all possible way from $\{l_{1},l_{2},\dots,l_{s}\}$ with at least once for each element of the set. The other positions of the hypermatrix are zeroFor a similar construction on uniform multi-hypergraph see ..

Let $G=(V,E)$ be a hypergraph in example 3.1. Here, the maximum cardinality of edges is $3$. The adjacency hypermatrix of $G$ is $\mathcal{A}_{G}=(a_{i_{1}i_{2}i_{3}})$, where $1\leq i_{1},i_{2},i_{3}\leq 5$. Here, $a_{111}=1,a_{233}=a_{232}=a_{223}=a_{323}=a_{332}=a_{322}=\frac{1}{3},a_{145}=a_{154}=a_{451}=a_{415}=a_{541}=a_{514}=\frac{1}{2}$, and the other elements of $\mathcal{A}_{G}$ are zero.

Let $G=(V,E)$ be a hypergraph. The degree, $d(v)$, of a vertex $v\in V$ is the number of edges consist of $v$.

Let $G=(V,E)$ be a hypergraph, where $V=\{v_{1},v_{2},\dots,v_{n}\}$ and $E=\{e_{1},e_{2},\dots,e_{k}\}$. Then, the degree of a vertex $v_{i}$ is given by

A hypergraph is called $k$-regular if every vertex has the same degree $k$.

Now, we discuss some spectral properties of $\mathcal{A}_{G}$ of a hypergraph $G$. Some of these properties are very similar as in general graph (i.e. for a $2$-uniform hypergraph).

Let $\mu$ be an $H$-eigenvalue of $\mathcal{A}_{G}$. Then $|\mu|\leq\Delta$, where $\Delta$ is the maximum degree of $G$.

Let $G$ be a hypergraph with $n$ vertices and $m.c.e(G)=m$. Let $\mu$ be an $H$-eigenvalue of $\mathcal{A}_{G}=(a_{i_{1}i_{2}\dots i_{m}})$ with an eigenvector $x=(x_{1},x_{2},\dots,x_{n})$. Let $x_{p}=max\{|x_{1}|,|x_{2}|,\dots,|x_{n}|\}$. Without loss of any generality we can assume that $x_{p}=1$. Now,

Thus, for a $k$-regular hypergraph the theorem (3.1) implies $|\mu|\leq k$.

Let $G=(V,E)$ be a $k$-regular hypergraph with $n$ vertices. Then, $\mathcal{A}_{G}=(a_{i_{1}i_{2}\dots i_{m}})$ has an $H$-eigenvalue $k$.

Let $G=(V,E)$ be a $k$-regular hypergraph with $n$ vertices. Then, $\mathcal{A}_{G}=(a_{i_{1}i_{2}\dots i_{m}})$ has a $Z$-eigenvalue $k(\frac{1}{\sqrt{n}})^{m-2}$.

Let $G$ be a hypergraph with $n$ vertices and maximum degree $\Delta$. Let $x=(x_{1},x_{2},\dots,x_{n})$ be a $Z$-eigenvector of $\mathcal{A}_{G}=(a_{i_{1}i_{2}\dots i_{m}})$ corresponding to an eigenvalue $\mu$. If x_{p}=max\big{\{}|x_{1}|,|x_{2}|,\dots,|x_{n}|\big{\}}, then $|\mu|\leq\frac{\Delta}{x_{p}}$.

The $Z$-eigenvalue equations of $\mathcal{A}_{G}$ for $\mu$ and $x$ are $Ax^{m-1}=\mu x$, and $\sum x_{i}^{2}=1$. Therefore, $|x_{i}|\leq 1$, for all $i=1,2,3,\dots,n$. Now,

which implies $|\mu||x_{j}|\leq d(j)\leq\Delta,\forall j=1,2,3,\dots,n$. Therefore, $|\mu|\leq\frac{\Delta}{x_{p}}$. ∎

A hypergraph $H=(V_{1},E_{1})$ is said to be a spanning subhypergraph of a hypergraph $G=(V,E)$, if $V=V_{1}$ and $E_{1}\subseteq E$.

Let $G=(V,E)$ be hypergraph. Let $H=(V^{{}^{\prime}},E^{{}^{\prime}})$ be a subhypergraph of $G$, such that, $m.c.e(G)=m.c.e(H)$ be even. Then, $\mu_{max}(H)\leq\mu_{max}(G)$, where $\mu_{max}$ is the highest $Z$-eigenvalue of the corresponding adjacency hypermatrix.

Let $|V|=n$, $|V^{{}^{\prime}}|=n^{{}^{\prime}}$ ($\leq n$) and $m.c.e(G)=m.c.e(H)=m.$ Now,

since each component of $x$ is nonnegative (by Perron-Frobenious theorem ) and the number of edges of $G$ is greater than or equal to the number of edges of $H$. Hence the proof. ∎

where the sum is over $r_{1},r_{2},\dots,r_{p}$ chosen in all possible way from $\{l_{1},l_{2},\dots,l_{s}\}$, such that, all $l_{j}(j\neq i)$ occur at least once. Whereas,

where the sum is over $r_{1},r_{2},\dots,r_{p}$ chosen in all possible way from $\{l_{1},l_{2},\dots,l_{s}\}$ with at least once for each element of the set.

The symmetric (adjacency) hypermatrix $\mathcal{A}_{G}$ of order $m$ and dimension $n$ uniquely defines a homogeneous polynomial of degree $m$ and in $n$ variables by

where $a_{e}^{G}=\frac{s}{\alpha}$, $\alpha=\sum_{k_{1},k_{2},\dots,k_{s}\geq 1,\sum k_{i}=m}\frac{m!}{k_{1}!k_{2}!\dots k_{s}!}$, and $s$ is the cardinality of the edge $e$.

Let $G$ and $H$ be two hypergraphs. The Cartesian product, $G\times H$, of $G$ and $H$ is defined by the vertex set $V(G\times H)=V(G)\times V(H)$ and the edge set E(G\times H)=\big{\{}\{v\}\times e:v\in V(G),e\in E(H)\big{\}}\bigcup\big{\{}e\times\{v\}:e\in E(G),v\in V(H)\big{\}}.

Let $G$ be a hypergraph with the vertex set $V=\{v_{1},v_{2},\dots,v_{n}\}$ and $m.c.e(G)=m$. For an edge $e=\{v_{l_{1}},v_{l_{2}},\dots,v_{l_{s}}\}$ and an integer $r\geq m$, the arrangement $(v_{p_{1}}v_{p_{2}}\dots v_{p_{r}})$ (where $p_{1},p_{2},\dots,p_{r}$ are chosen in all possible way from $\{l_{1},l_{2},\dots,l_{s}\}$ with at least once for each element of the set) represents the edge $e$ in order $r$.

Let $G=(V,E)$ where $V=\{1,2,3,4,5\}$ and E=\big{\{}\{1,2,3\},\{2,3,5\},\{1,3,4,5\}\big{\}}, then the arrangement $(12233)$ represents the edge $\{1,2,3\}$ in order 5. $(12123)$ is also a representation of the edge $\{1,2,3\}$ in order five, whereas, $(111123)$ represents the edge $\{1,2,3\}$ in 6 order.

Let $G=(V,E)$ be a hypergraph with $m.c.e(G)=m$ and $E_{i}=\{e\in E:v_{i}\in e\}$. Now, the $H$-eigenvalue equation for $\mathcal{A}_{G}$ becomes

Let $G$ and $H$ be two hypergraphs with $m.c.e(G)=m.c.e(H)$. If $\lambda$ and $\mu$ are $H$-eigenvalue for $G$ and $H$, respectively, then $\lambda+\mu$ is an $H$-eigenvalue for $G\times H$.

Hence the proofFor similar proof on uniform hypergraph see .. ∎

Let $A$ and $B$ be two symmetric hypermatrices of order $m$ and dimension $n$, where $m$ is even. Then $\lambda_{max}(A+B)\leq\lambda_{max}(A)+\lambda_{max}(B)$, where $\lambda_{max}(A)$ denotes the largest $Z$-eigenvalue of $A$.

Let $G=(V,E)$ be a hypergraph with the vertex set $V=\{v_{1},v_{2},\dots,v_{n}\}$ and $m=max\{|e_{i}|:e_{i}\in E\}$. We partition the edge set $E$ as, $E=E_{1}\cup E_{2}\cup\dots\cup E_{m}$, where $E_{i}$ contains all the edges of the cardinality $i$ and construct a hypergraph $G_{i}=(V,E_{i})$, for a nonempty $E_{i}$.

Define the adjacency hypermatrix of $G_{i}$ in $m$ $(>i)$ -order by an $n$ dimensional $m$ order hypermatrix

such that, for any $e=\{v_{l_{1}},v_{l_{2}},\dots,v_{l_{i}}\}\in E_{i}$,

and $p_{1},p_{2},\dots,p_{m}$ are chosen in all possible way from $\{l_{1},l_{2},\dots,l_{i}\}$ with at least once for each element of the set. The other positions of $\mathcal{A}_{G_{i}}^{m}$ are zero.

Thus, we can represent a hypergraph $G$, with $m.c.e(G)=s$, in higher order $m>s$ by the hypermatrix $\mathcal{A}_{G}^{m}$. Clearly, all the eigenvalue equations show that the eigenvalues of $\mathcal{A}_{G}^{m_{1}}$ and $\mathcal{A}_{G}^{m_{2}}$ are not equal for $m_{1}\neq m_{2}$.

Let $G=(V,E)$ be a hypergraph and $m.c.e(G)=m$ be even. Then $\lambda_{max}(\mathcal{A}_{G})\leq\sum_{i=1}^{m}\lambda_{max}(\mathcal{A}_{G_{i}}^{m})$, where $\lambda_{max}({A})$ is the largest $Z$-eigenvalue of $A$.

Since $\mathcal{A}_{G}=\sum_{i=1}^{m}\mathcal{A}_{G_{i}}^{m}$, the proof follows from the lemma (3.1). ∎

Moreover, the theorem (3.7) implies $\lambda_{max}(\mathcal{A}_{G})\leq\sum_{i=1}^{m}n_{i}\lambda_{max}(\mathcal{A}_{i}^{m})$, where $n_{i}$ is the number of edges of cardinality $i$ and $\mathcal{A}_{i}^{m}$ is the adjacency hypermatrix in $m$-order of a hypergraph contains a single edge of cardinality $i$.

2 Laplacian hypermatrix and eigenvalues

Let $G=(V,E)$ be a (general) hypergraph without any isolated vertex where $V=\{v_{1},v_{2},\dots,v_{n}\}$ and $E=\{e_{1},e_{2},\dots,e_{k}\}$. Let $m.c.e(G)=m$. We define the Laplacian hypermatrix, $L_{G}$, of $G=(V,E)$ as $L_{G}=D_{G}-\mathcal{A}_{G}=(l_{i_{1}i_{2}\dots i_{m}}),1\leq i_{1},i_{2},\dots,i_{m}\leq n,$ where $D_{G}=(d_{i_{1}i_{2}\dots i_{m}})$ is the $m$ order $n$ dimensional diagonal hypermatrix with $d_{ii\dots i}=d(v_{i})$ and others are zero. The signless Laplacian of $G$ is defined as $L_{G}=D_{G}+\mathcal{A}_{G}.$

Let $G=(V,E)$ be a hypergraph with $m.c.e(G)=m$. For any edge $e=\{v_{l_{1}},v_{l_{2}},\dots,v_{l_{s}}\}$, we define a homogeneous polynomial of degree $m$ and in $n$ variables by

$x_{m}^{e}$ is the sum of all possible terms, $x_{i_{1}}^{k_{1}}x_{i_{2}}^{k_{2}}\dots x_{i_{s}}^{k_{s}}$ (where $\sum k_{i}=m$ and $k_{i}\geq 1$)

with some natural coefficient. Now, by applying AM-GM inequality on $k_{1}x_{i_{1}}^{m},k_{2}x_{i_{2}}^{m},\dots,k_{s}x_{i_{s}}^{m}$ we get

If we apply (1) for each term of $x_{m}^{e}$ and take the sum, we get

Let $G=(V,E)$ be a general hypergraph. Let $L=(l_{i_{1}i_{2}\dots i_{m}})\text{ where }1\leq i_{1},i_{2},\dots,i_{m}\leq n,$ be the Laplacian hypermatrix of $G$. Then $0\leq\lambda\leq 2\Delta,$ where $\lambda$ is an $H$-eigenvalue of $L$.

i.e., $|\lambda|\leq 2\Delta.$ Thus $0\leq\lambda\leq 2\Delta$. ∎

Let $G=(V,E)$ be a general hypergraph with $m.c.e(G)=m\geq 3$. Let $L$ be the Laplacian hypermatrix of $G$. Then

$\Delta$ is the largest $H^{+}$-eigenvalue of $L$.

Thus, $\lambda\leq d(v_{j})-d(v_{j})=0$. Hence $\lambda=0$.

Suppose $\lambda$ is an $H^{+}$-eigenvalue with non-negative $H^{+}$-eigenvector, $x$ of $L$. Assume that $x_{j}>0$. Now, we have

Therefore $\lambda\leq d(v_{j})\leq\Delta$. Thus, $\Delta$ is the largest $H^{+}$-eigenvalue of $L$.

It is clear from the eigenvalue equation.

The general hypergraph $G=(V,E)$ with $m.c.e(G)\geq 3$ is connected if and only if $\alpha(G)>0.$

Therefore, $x_{i}=x_{k}$ as long as $i$ and $k$ belong to same edge. Thus, $x_{i}=x_{k}$ as long as $i$ and $k$ are in same component of $G$. Since $y_{j}=0$, we have, $j$ and $k$ are in different components of $G$. Hence, $G$ is not connected. This proves the theorem. ∎

3 Normalized Laplacian hypermatrix and eigenvalues

Now, we define normalized Laplacian hypermatrix for a general hypergraph. For any graph, there are two ways to construct normalized Laplacian matrix (see and for details)These two matrices are similar, i.e., they have same eigenvalues.. Motivated by these two similar constructions, here, we also define the normalized Laplacian hypermatrix in two different ways and show that they are cospectral. The first definition is similar to the normalized Laplacian matrix defined in .

Let $G=(V,E)$ be a general hypergraph without any isolated vertex where $V=\{v_{1},v_{2},\dots,v_{n}\}$ and $E=\{e_{1},e_{2},\dots,e_{k}\}$. Let $m.c.e(G)=m$. The normalized Laplacian hypermatrix $\mathcal{L}=(l_{i_{1}i_{2}\dots i_{m}})$, which is an $n$-dimensional $m$-th order hypermatrix, is defined as: for any edge $e=\{v_{l_{1}},v_{l_{2}},\dots,v_{l_{s}}\}\in E$ of cardinality $s\leq m$,

and $p_{1},p_{2},\dots,p_{m}$ are chosen in all possible way from $\{l_{1},l_{2},\dots,l_{s}\}$, such that, all $l_{j}$ occur at least once. All the diagonal entries are 1 and the rest are zero.

Clearly, the hypermatrix $\mathit{A}=\mathcal{I}-\mathcal{L}$, which is known as normalized adjacency hypermatrix, is a stochastic tensor, that is, $\mathit{A}$ is non-negative and $\sum_{i_{2},\dots,i_{m}=1}^{n}a_{ii_{2}\dots i_{m}}=1$, where $a_{i_{1}i_{2}\dots i_{m}}$ is the $(i_{1},i_{2},\dots,i_{m})$-th entry of $\mathit{A}$. The different properties of a stochastic tensor are discussed in and which can be used to study the hypermatrices $\mathit{A}$ and $\mathcal{L}$. Now, we define the normalized Laplacian hypermatrix of a general hypergraph as it is defined for a graph in .

Let $G=(V,E)$ be a general hypergraph without any isolated vertex, where $V=\{v_{1},v_{2},\dots,v_{n}\}$ and $E=\{e_{1},e_{2},\dots,e_{k}\}$. Let $m.c.e(G)=m$. The normalized Laplacian hypermatrix $\mathfrak{L}=(l_{i_{1}i_{2}\dots i_{m}})$, which is an $n$-dimension $m$-th order symmetric hypermatrix, is defined as: for any edge $e=\{v_{l_{1}},v_{l_{2}},\dots,v_{l_{s}}\}\in E$ of cardinality $s\leq m$,

and $p_{1},p_{2},\dots,p_{m}$ chosen in all possible way from $\{l_{1},l_{2},\dots,l_{s}\}$ with at least once for each element of the set. The diagonal entries of $\mathfrak{L}$ are $1$ and the rest of the positions are zero.

$\mathcal{L}$ and $\mathfrak{L}$ are co-spectral.

In the lemma (2.1) choose a diagonal matrix $D=(d_{ij})_{n\times n}$ where $d_{ii}=(d(v_{i}))^{1/m}$. ∎

Let $G=(V,E)$ be a general hypergraph. Let $\mathcal{L}$, $\mathit{A}$ be the normalized Laplacian and normalized adjacency hypermatrices of $G$, respectively. If $G$ has at least one edge, then $\lambda\in\sigma(\mathit{A})$ if and only if $(1-\lambda)\in\sigma(\mathcal{L})$, otherwise, $\sigma(\mathit{A})=\sigma(\mathcal{L})={0}$, where $\sigma(\mathcal{L})$ denotes the spectrum of $\mathcal{L}$.

Since, $\mathcal{L}=\mathcal{I}-\mathit{A}$ and $\lambda$ is the eigenvalue of $\mathit{A}$ iff $\det(\mathit{A}-\lambda\mathcal{I})=0,$ thus, $\det(\mathcal{L}-(1-\lambda)\mathcal{I})=0$ implies $(1-\lambda)\in\sigma(\mathcal{L})$. ∎

Let $G=(V,E)$ be a general hypergraph. Let $\mathcal{L}=(l_{i_{1}i_{2}\dots i_{m}})\text{ where }1\leq i_{1},i_{2},\dots,i_{m}\leq n,$ and $A$ be the normalized Laplacian and normalized adjacency hypermatrices of $G$, respectively, then

1 is the largest $H^{+}$-eigenvalue of $\mathcal{L}$.

is the unique $H^{++}$-eigenvalue of $\mathcal{L}$ .

Since $A$ is stochastic tensor, it is obvious that the spectral radius of $A$ is 1. Moreover, $(1,1,\dots,1)$ is an eigenvector with eigenvalue 1.

We know that spectral radius of $A$ is 1 and $\mathcal{L}=\mathcal{I}-A$. By theorem (3.12) $\lambda\in\sigma(\mathit{A})$ if and only if $(1-\lambda)\in\sigma(\mathcal{L})$. Since 1 is an eigenvalue of $A$, thus, $\lambda\geq 0$. Again, using theorem ($6(a)$) of we have

This implies $|\lambda(\mathcal{L})|\leq 2.$ Thus, we have $0\leq\lambda(\mathcal{L})\leq 2$.

Suppose that $\lambda$ is an $H^{+}$-eigenvalue with non-negative $H^{+}$-eigenvector, $x$ of $\mathcal{L}$. Assume that $x_{j}>0$. Now, we have

Hence, $\lambda x_{j}^{m-1}\leq x_{j}^{m-1}$ implies $\lambda\leq 1$. Thus, 1 is the largest $H^{+}$-eigenvalue of $\mathcal{L}$.

Let $x$ is an $H^{++}$-eigenvector of $\mathcal{L}$ with eigenvalue $\lambda$. From the part (ii) of this theorem we have $\lambda\geq 0.$ Suppose $x_{j}=\underset{i}{min}\{x_{i}\}$. Now,

Thus $\lambda\leq 1-1=0$. Hence $\lambda=0$.

Let $G=(V,E)$ be a general hypergraph and $m.c.e(G)=m$. Let $\mathcal{L}$ be the normalized Laplacian hypermatrix of $G$ of order m and dimension n. Let $m(\lambda)$ be the algebraic multiplicity of $\lambda\in\sigma(\mathcal{L})$, then $\sum_{\lambda\in\sigma(\mathcal{L})}m(\lambda)\lambda=n(m-1)^{n-1}.$

Hence, we have $\sum_{\lambda\in\sigma(\mathcal{L})}m(\lambda)\lambda=n(m-1)^{n-1}.$ ∎

Let $G=(V,E)$ be a general hypergraph and $A$ be any connectivity hypermatrix of $G$. If $G$ has $r\geq 1$ connected components, $G_{1},G_{2},\dots,G_{r}$, such that, $|V(G_{i})|=n_{i}>1$ and $m.c.e(G_{i})=m.c.e(G)$ for each $i\in\{1,2,\dots,r\}$. Then, as sets, $\sigma(A)=\sigma(A_{1})\cup\sigma(A_{2})\cup\dots\cup\sigma(A_{r}),$ where $A_{i}$ is the connectivity hypermatrix of $G_{i}$.

where $\phi_{A}(\lambda)$ is the characteristic polynomial of the tensor $A$. Therefore, $\sigma(A)=\sigma(A_{1})\cup\sigma(A_{2})\cup\dots\cup\sigma(A_{r})$. ∎

Discussion and conclusion

Here, we propose a mathematical framework to construct connectivity matrices for a general hypergraph and also study the eigenvalues of adjacency hypermatrix, Laplacian hypermatrix, normalized Laplacian hypermatrix. This connectivity hypermatrix reconstruction can be used for further development of spectral hypergraph theory in many aspects, but, this may not be quite useful to study dynamics on hypergraphs.

Acknowledgements

The authors are thankful to Mithun Mukherjee and Swarnendu Datta for fruitful discussions. Financial support from Council of Scientific and Industrial Research, India, Grant no-09/921(0113)/2014-EMR-I is sincerely acknowledged by Bibhash Mondal.

References