The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph

The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph

Shenglong Hu ,     Liqun Qi ,      Jinshan Xie Email: Tim.Hu@connect.polyu.hk. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.Email: maqilq@polyu.edu.hk. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. This author’s work was supported by the Hong Kong Research Grant Council (Grant No. PolyU 501909, 502510, 502111 and 501212).Email: jinshan0623@sina.com. School of Mathematics and Computer Science, Longyan University, Longyan, Fujian, China.
July 1, 2019
Abstract

In this paper, we show that the largest Laplacian H-eigenvalue of a -uniform nontrivial hypergraph is strictly larger than the maximum degree when is even. A tight lower bound for this eigenvalue is given. For a connected even-uniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when is odd, in certain cases the largest Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower bound. On the other hand, tight upper and lower bounds for the largest signless Laplacian H-eigenvalue of a -uniform connected hypergraph are given. For a connected -uniform hypergraph, the upper (respectively lower) bound of the largest signless Laplacian H-eigenvalue is achieved if and only if it is a complete hypergraph (respectively a hyperstar). The largest Laplacian H-eigenvalue is always less than or equal to the largest signless Laplacian H-eigenvalue. When the hypergraph is connected, the equality holds here if and only if is even and the hypergraph is odd-bipartite.

Key words:   Tensor, H-eigenvalue, hypergraph, Laplacian, signless Laplacian

MSC (2010):   05C65; 15A18

1 Introduction

In this paper, we study the largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph. The largest Laplacian and signless Laplacian H-eigenvalues refer to respectively the largest H-eigenvalue of the Laplacian tensor and the largest H-eigenvalue of the signless Laplacian tensor. This work is motivated by the classic results for graphs [4, 2, 6, 25, 24]. Please refer to [14, 9, 5, 17, 15, 12, 20, 19, 22, 10, 3, 8, 13, 16, 18, 21, 23] for recent developments on spectral hypergraph theory and the essential tools from spectral theory of nonnegative tensors.

This work is a companion of the recent study on the eigenvectors of the zero Laplacian and signless Laplacian eigenvalues of a uniform hypergraph by Hu and Qi [11]. For the literature on the Laplacian-type tensors for a uniform hypergraph, which becomes an active research frontier in spectral hypergraph theory, please refer to [9, 12, 22, 17, 10, 23, 11] and references therein. Among others, Qi [17], and Hu and Qi [10] respectively systematically studied the Laplacian and signless Laplacian tensors, and the Laplacian of a uniform hypergraph. These three notions of Laplacian-type tensors are more natural and simpler than those in the literature.

The rest of this paper is organized as follows. Some definitions on eigenvalues of tensors and uniform hypergraphs are presented in the next section. The class of hyperstars is introduced. We discuss in Section 3 the largest Laplacian H-eigenvalue of a -uniform hypergraph. We show that when is even, the largest Laplacian H-eigenvalue has a tight lower bound that is strictly larger than the maximum degree. Extreme hypergraphs in this case are characterized, which are the hyperstars. When is odd, a tight lower bound is exactly the maximum degree. However, we are not able to characterize the extreme hypergraphs in this case. Then we discuss the largest signless Laplacian H-eigenvalue in Section 4. Tight lower and upper bounds for the largest signless Laplacian H-eigenvalue of a connected hypergraph are given. Extreme hypergraphs are characterized as well. For the lower bound, the extreme hypergraphs are hyperstars; and for the upper bound, the extreme hypergraphs are complete hypergraphs. The relationship between the largest Laplacian H-eigenvalue and the largest signless Laplacian H-eigenvalue is discussed in Section 5. The largest Laplacian H-eigenvalue is always less than or equal to the largest signless Laplacian H-eigenvalue. When the hypergraph is connected, the equality holds here if and only if is even and the hypergraph is odd-bipartite. This result can help to find the largest Laplacian H-eigenvalue of an even-uniform hypercycle. Some final remarks are made in the last section.

2 Preliminaries

Some definitions of eigenvalues of tensors and uniform hypergraphs are presented in this section.

2.1 Eigenvalues of Tensors

In this subsection, some basic definitions on eigenvalues of tensors are reviewed. For comprehensive references, see [16, 8] and references therein. Especially, for spectral hypergraph theory oriented facts on eigenvalues of tensors, please see [17, 10].

Let be the field of real numbers and the -dimensional real space. denotes the nonnegative orthant of . For integers and , a real tensor of order and dimension refers to a multiway array (also called hypermatrix) with entries such that for all and . Tensors are always referred to -th order real tensors in this paper, and the dimensions will be clear from the content. Given a vector , is defined as an -dimensional vector such that its -th element being for all . Let be the identity tensor of appropriate dimension, e.g., if and only if , and zero otherwise when the dimension is . The following definition was introduced by Qi [16].

Definition 2.1

Let be a -th order -dimensional real tensor. For some , if polynomial system has a solution , then is called an H-eigenvalue and an H-eigenvector.

It is seen that H-eigenvalues are real numbers [16]. By [8, 16], we have that the number of H-eigenvalues of a real tensor is finite. By [17], we have that all the tensors considered in this paper have at least one H-eigenvalue. Hence, we can denote by (respectively ) as the largest (respectively smallest) H-eigenvalue of a real tensor .

For a subset , we denoted by its cardinality, and its support.

2.2 Uniform Hypergraphs

In this subsection, we present some essential concepts of uniform hypergraphs which will be used in the sequel. Please refer to [1, 4, 2, 10, 17] for comprehensive references.

In this paper, unless stated otherwise, a hypergraph means an undirected simple -uniform hypergraph with vertex set , which is labeled as , and edge set . By -uniformity, we mean that for every edge , the cardinality of is equal to . Throughout this paper, and . Moreover, since the trivial hypergraph (i.e., ) is of less interest, we consider only hypergraphs having at least one edge (i.e., nontrivial) in this paper.

For a subset , we denoted by the set of edges . For a vertex , we simplify as . It is the set of edges containing the vertex , i.e., . The cardinality of the set is defined as the degree of the vertex , which is denoted by . Two different vertices and are connected to each other (or the pair and is connected), if there is a sequence of edges such that , and for all . A hypergraph is called connected, if every pair of different vertices of is connected. Let , the hypergraph with vertex set and edge set is called the sub-hypergraph of induced by . We will denote it by . A hypergraph is regular if . A hypergraph is complete if consists of all the possible edges. In this case, is regular, and moreover . In the sequel, unless stated otherwise, all the notations introduced above are reserved for the specific meanings.

For the sake of simplicity, we mainly consider connected hypergraphs in the subsequent analysis. By the techniques in [17, 10], the conclusions on connected hypergraphs can be easily generalized to general hypergraphs.

The following definition for the Laplacian tensor and signless Laplacian tensor was proposed by Qi [17].

Definition 2.2

Let be a -uniform hypergraph. The adjacency tensor of is defined as the -th order -dimensional tensor whose -entry is:

Let be a -th order -dimensional diagonal tensor with its diagonal element being , the degree of vertex , for all . Then is the Laplacian tensor of the hypergraph , and is the signless Laplacian tensor of the hypergraph .

In the following, we introduce the class of hyperstars.

Definition 2.3

Let be a -uniform hypergraph. If there is a disjoint partition of the vertex set as such that and , and , then is called a hyperstar. The degree of the vertex in , which is called the heart, is the size of the hyperstar. The edges of are leaves, and the vertices other than the heart are vertices of leaves.

It is an obvious fact that, with a possible renumbering of the vertices, all the hyperstars with the same size are identical. Moreover, by Definition 2.1, we see that the process of renumbering does not change the H-eigenvalues of either the Laplacian tensor or the signless Laplacian tensor of the hyperstar. The trivial hyperstar is the one edge hypergraph, its spectrum is very clear [5]. In the sequel, unless stated otherwise, a hyperstar is referred to a hyperstar having size . For a vertex other than the heart, the leaf containing is denoted by . An example of a hyperstar is given in Figure 1.

Figure 1: An example of a -uniform hyperstar of size . An edge is pictured as a closed curve with the containing solid disks the vertices in that edge. Different edges are in different curves with different colors. The red (also in dashed margin) disk represents the heart.

The notions of odd-bipartite and even-bipartite even-uniform hypergraphs are introduced in [11].

Definition 2.4

Let be even and be a -uniform hypergraph. It is called odd-bipartite if either it is trivial (i.e., ) or there is a disjoint partition of the vertex set as such that and every edge in intersects with exactly an odd number of vertices.

An example of an odd-bipartite hypergraph is given in Figure 2.

Figure 2: An example of an odd-bipartite -uniform hypergraph. The bipartition is clear from the different colors (also the dashed margins from the solid ones) of the disks.

3 The Largest Laplacian H-Eigenvalue

This section presents some basic facts about the largest Laplacian H-eigenvalue of a uniform hypergraph. We start the discussion on the class of hyperstars.

3.1 Hyperstars

Some properties of hyperstars are given in this subsection.

The next proposition is a direct consequence of Definition 2.3.

Proposition 3.1

Let be a hyperstar of size . Then except for one vertex with , we have for the others.

By Theorem 4 of [17], we have the following lemma.

Lemma 3.1

Let be a -uniform hypergraph with its maximum degree and be its Laplacian tensor. Then .

When is even and is a hyperstar, Lemma 3.1 can be strengthened as in the next proposition.

Proposition 3.2

Let be even and be a hyperstar of size and be its Laplacian tensor. Then .

Proof. Suppose, without loss of generality, that . Let be a nonzero vector such that , and . Then, we see that

and for

Thus, if is an H-eigenvector of corresponding to an H-eigenvalue , then we must have

Hence,

Let . We have that

Consequently, does have a root in the interval . Hence has an H-eigenvalue . The result follows.

The next lemma characterizes H-eigenvectors of the Laplacian tensor of a hyperstar corresponding to an H-eigenvalue which is not one .

Lemma 3.2

Let be a hyperstar of size and be an H-eigenvector of the Laplacian tensor of corresponding to a nonzero H-eigenvalue other than one. If for some vertex of a leaf (other than the heart), then for all the vertices in the leaf containing and other than the heart. Moreover, in this situation, if is the heart, then .

Proof. Suppose that the H-eigenvalue is . By the definition of eigenvalues, we have that for the vertex other than the heart and the vertex ,

Since , we must have that .

With a similar proof, we get the other conclusion by contradiction, since for all vertices of leaves and .

The next lemma characterizes the H-eigenvectors of the Laplacian tensor of a hyperstar corresponding to the largest Laplacian H-eigenvalue.

Lemma 3.3

Let be a hyperstar of size . Then there is an H-eigenvector of the Laplacian tensor of corresponding to satisfying that is a constant for and being not the heart.

Proof. Suppose that is an H-eigenvector of corresponding to . Without loss of generality, let be the heart and hence . Note that, by Lemma 3.1, we have that . By Lemma 3.2, without loss of generality, we can assume that and . In the following, we construct an H-eigenvector corresponding to from such that .

(I). We first prove that for every leaf , is a constant for all .

For an arbitrary but fixed leaf , suppose that and . If , then we are done. In the following, suppose on the contrary that . Then, we have

By the definitions of and , we have . On the other hand, we have . Hence, a contradiction is derived. Consequently, for every leaf , is a constant for all .

(II). We next show that all the numbers in this set

are of the same sign.

When is even, suppose that for some . Then

(1)

Thus, an odd number of vertices in takes negative values. By (1), we must have that there exists some such that for every . Otherwise, , together with , would lead to a contradiction. Hence, all the numbers in this set

are negative.

When is odd, suppose that for some . Then

(2)

Thus, an positive even number of vertices in takes negative values. Thus, if there is some such that , then

Since , we have and . Hence, . A contradiction is derived. By (2), we must have that there exists some such that for every . Consequently, for all . Hence, all the numbers in this set

are positive.

(III.) We construct the desired vector .

If the product is a constant for every leaf , then take and we are done. In the following, suppose on the contrary that the set

takes more than one numbers. Let be the vector such that

and . Note that , since for all and . Then

For any with for some , we have

By Definition 2.1, is an H-eigenvector of corresponding to with the requirement. The result follows.

The next corollary follows directly from the proof of Lemma 3.3.

Corollary 3.1

Let be odd and be a hyperstar of size . If is an H-eigenvector of the Laplacian tensor of corresponding to , then is a constant for and being not the heart. Moreover, whenever contains a vertex other than the heart, the signs of the heart and the vertices of leaves in are opposite.

However, in Section 3.3, we will show that is a singleton which is the heart.

The next lemma is useful, which follows from a similar proof of [16, Theorem 5].

Lemma 3.4

Let be even and be a -uniform hypergraph. Let be the Laplacian tensor of . Then

(3)

The next lemma is an analogue of Corollary 3.1 for being even.

Lemma 3.5

Let be even and be a hyperstar of size . Then there is an H-eigenvector of the Laplacian tensor of satisfying that is a constant for and being not the heart.

Proof. In the proof of Lemma 3.3, is required only to guarantee . While, when is even, by Proposition 3.2, whenever . Hence, there is an H-eigenvector of the Laplacian tensor of corresponding to satisfying that is a constant for and being not the heart.

Suppose, without loss of generality, that is the heart. By Lemma 3.2, without loss of generality, suppose that . If , then let , and otherwise let .

Suppose that for some other than . Then

Thus, a positive even number of vertices in other than takes negative values. Hence, all the values in this set

are positive. Let such that and for the others. We have that if , then

and if , then

Here, the second equality follows from the fact that in this situation. Moreover,

Consequently, is the desired H-eigenvector.

The next theorem gives the largest Laplacian H-eigenvalue of a hyperstar for being even.

Theorem 3.1

Let be even and be a hyperstar of size . Let be the Laplacian tensor of . Then is the unique real root of the equation in the interval .

Proof. By Lemma 3.5, there is an H-eigenvector of the Laplacian tensor of satisfying that is a constant for and being not the heart. By the proof for Lemma 3.2, we have that is the largest real root of the equation . Here is the size of the sub-hyperstar of .

Let . Then, . Hence, is strictly decreasing in the interval . Moreover, . Consequently, has a unique real root in the interval which is the maximum. Thus, by Proposition 3.2, we must have . The result follows.

The next corollary is a direct consequence of Theorem 3.1.

Corollary 3.2

Let and be two hyperstars of size and , respectively. Let and be the Laplacian tensors of and respectively. If , then .

When is even, the proofs of Lemmas 3.3 and 3.5, and Theorem 3.1 actually imply the next corollary.

Corollary 3.3

Let be even and be a hyperstar of size . If is an H-eigenvector of the Laplacian tensor of corresponding to , then . Hence, there is an H-eigenvector of the Laplacian tensor of corresponding to satisfying that is a constant for all the vertices other than the heart.

3.2 Even-Uniform Hypergraphs

In this subsection, we present a tight lower bound for the largest Laplacian H-eigenvalue and characterize the extreme hypergraphs when is even.

The next theorem gives the lower bound, which is tight by Theorem 3.1.

Theorem 3.2

Let be even and be a -uniform hypergraph with the maximum degree being . Let be the Laplacian tensor of . Then is not smaller than the unique real root of the equation in the interval .

Proof. Suppose that , the maximum degree. Let be a -uniform hypergraph such that and consisting of the vertex and the vertices which share an edge with . Let be the Laplacian tensor of . We claim that .

Suppose that and is an H-eigenvector of corresponding to the H-eigenvalue such that . Suppose, without loss of generality, that , and the degree of vertex in the hypergraph is . Let such that

(4)

Obviously, . Moreover,

Here the inequality follows from the fact that by the arithmetic-geometric mean inequality. Thus, by the characterization (3) (Lemma 3.4), we get the conclusion since .

For the hypergraph , we define a new hypergraph by renumbering the vertices in the following way: fix the vertex , and for every edge , number the rest vertices as . Let be the -uniform hypergraph with and . It is easy to see that is a hyperstar with size and the heart being (Definition 2.3). Let be an H-eigenvector of the Laplacian tensor of corresponding to . Suppose that . By Corollary 3.3, we can choose a such that is a constant other than which corresponds to the heart. Let be defined as being the constant for all and . Then, by a direct computation, we see that

Moreover, . By (3) and the fact that (Theorem 3.1), we see that

(6)

Consequently, . By Theorem 3.1, is the unique real root of the equation in the interval . Consequently, is no smaller than the unique real root of the equation in the interval .

By the proof of Theorem 3.2, the next theorem follows immediately.

Theorem 3.3

Let be even, and and be two -uniform hypergraphs. Suppose that and be the Laplacian tensors of and respectively. If and , then .

The next lemma helps us to characterize the extreme hypergraphs with respect to the lower bound of the largest Laplacian H-eigenvalue.

Lemma 3.6

Let be even and be a hyperstar of size . Then there is an H-eigenvector of the Laplacian tensor of satisfying that exactly two vertices other than the heart in every edge takes negative values.

Proof. Suppose, without loss of generality, that is the heart. By Corollary 3.3, there is an H-eigenvector of corresponding to such that is a constant for the vertices other than the heart. By Theorem 3.1, we have that this constant is nonzero. If , then let , and otherwise let . We have that is an H-eigenvector of corresponding to .

Let . We set , and for every edge arbitrarily two chosen we set , and for the others . Then, by a direct computation, we can conclude that is an H-eigenvector of corresponding to .

The next theorem is the main result of this subsection, which characterizes the extreme hypergraphs with respect to the lower bound of the largest Laplacian H-eigenvalue.

Theorem 3.4

Let be even and be a -uniform connected hypergraph with the maximum degree being . Let be the Laplacian tensor of . Then is equal to the unique real root of the equation in the interval if and only if is a hyperstar.

Proof. By Theorem 3.1, only necessity needs a proof. In the following, suppose that is equal to the unique real root of the equation in the interval . Suppose that as before.

Define and as in Theorem 3.2. Actually, let be the -uniform hypergraph such that and consisting of the vertex and the vertices which share an edge with . Let be the Laplacian tensor of . Fix the vertex , and for every edge , number the rest vertices as . Let be the -uniform hypergraph such that and .

With the same proof as in Theorem 3.2, by Lemma 3.4, we have that inequality in (6) is an equality if and only if . Since otherwise , which together with and (3) implies that . Hence, if is equal to the unique real root of the equation in the interval , then is a hyperstar. In this situation, the inequality in (3.2) is an equality if and only if . The sufficiency is clear.

For the necessity, suppose that . Then there is an edge

  • either containing both vertices in and vertices in , since is connected,

  • or containing only vertices in .

For the case (i), it is easy to get a contradiction since . Note that this situation happens if and only if . Then, in the following we assume that that . For the case (ii), we must have that there are edges in such that for all . By Lemma 3.6, let be an H-eigenvector of the Laplacian tensor of satisfying that exactly two vertices other than the heart in every edge takes negative values. Moreover, we can normalize such that . Since , by (4), we have . Consequently, by Lemma 3.4, we have

If , then we get a contradiction since is equal to the unique real root of the equation in the interval . In the following, we assume that . We have two cases:

  • or for all ,

  • for some and for some .

Note that for all . For an arbitrary but fixed , define .

(I). If , then we choose an such that , and . Since is even, such an exists. It is a direct computation to see that such that , , and for the others is still an H-eigenvector of