The Largest Laplacian and Signless Laplacian HEigenvalues of a Uniform Hypergraph
Abstract
In this paper, we show that the largest Laplacian Heigenvalue 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 evenuniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when is odd, in certain cases the largest Laplacian Heigenvalue 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 Heigenvalue of a uniform connected hypergraph are given. For a connected uniform hypergraph, the upper (respectively lower) bound of the largest signless Laplacian Heigenvalue is achieved if and only if it is a complete hypergraph (respectively a hyperstar). The largest Laplacian Heigenvalue is always less than or equal to the largest signless Laplacian Heigenvalue. When the hypergraph is connected, the equality holds here if and only if is even and the hypergraph is oddbipartite.
Key words: Tensor, Heigenvalue, hypergraph, Laplacian, signless Laplacian
MSC (2010): 05C65; 15A18
1 Introduction
In this paper, we study the largest Laplacian and signless Laplacian Heigenvalues of a uniform hypergraph. The largest Laplacian and signless Laplacian Heigenvalues refer to respectively the largest Heigenvalue of the Laplacian tensor and the largest Heigenvalue 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 Laplaciantype 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 Laplaciantype 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 Heigenvalue of a uniform hypergraph. We show that when is even, the largest Laplacian Heigenvalue 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 Heigenvalue in Section 4. Tight lower and upper bounds for the largest signless Laplacian Heigenvalue 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 Heigenvalue and the largest signless Laplacian Heigenvalue is discussed in Section 5. The largest Laplacian Heigenvalue is always less than or equal to the largest signless Laplacian Heigenvalue. When the hypergraph is connected, the equality holds here if and only if is even and the hypergraph is oddbipartite. This result can help to find the largest Laplacian Heigenvalue of an evenuniform 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 Heigenvalue and an Heigenvector.
It is seen that Heigenvalues are real numbers [16]. By [8, 16], we have that the number of Heigenvalues of a real tensor is finite. By [17], we have that all the tensors considered in this paper have at least one Heigenvalue. Hence, we can denote by (respectively ) as the largest (respectively smallest) Heigenvalue 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 subhypergraph 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 Heigenvalues 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.
The notions of oddbipartite and evenbipartite evenuniform hypergraphs are introduced in [11].
Definition 2.4
Let be even and be a uniform hypergraph. It is called oddbipartite 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 oddbipartite hypergraph is given in Figure 2.
3 The Largest Laplacian HEigenvalue
This section presents some basic facts about the largest Laplacian Heigenvalue 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 Heigenvector of corresponding to an Heigenvalue , then we must have
Hence,
Let . We have that
Consequently, does have a root in the interval . Hence has an Heigenvalue . The result follows.
The next lemma characterizes Heigenvectors of the Laplacian tensor of a hyperstar corresponding to an Heigenvalue which is not one .
Lemma 3.2
Let be a hyperstar of size and be an Heigenvector of the Laplacian tensor of corresponding to a nonzero Heigenvalue 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 Heigenvalue 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 Heigenvectors of the Laplacian tensor of a hyperstar corresponding to the largest Laplacian Heigenvalue.
Lemma 3.3
Let be a hyperstar of size . Then there is an Heigenvector of the Laplacian tensor of corresponding to satisfying that is a constant for and being not the heart.
Proof. Suppose that is an Heigenvector 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 Heigenvector 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 Heigenvector 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 Heigenvector 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 Heigenvector 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 Heigenvector 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 Heigenvector.
The next theorem gives the largest Laplacian Heigenvalue 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 Heigenvector 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 subhyperstar 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 .
Corollary 3.3
Let be even and be a hyperstar of size . If is an Heigenvector of the Laplacian tensor of corresponding to , then . Hence, there is an Heigenvector of the Laplacian tensor of corresponding to satisfying that is a constant for all the vertices other than the heart.
3.2 EvenUniform Hypergraphs
In this subsection, we present a tight lower bound for the largest Laplacian Heigenvalue 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 Heigenvector of corresponding to the Heigenvalue 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 arithmeticgeometric 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 Heigenvector 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 Heigenvalue.
Lemma 3.6
Let be even and be a hyperstar of size . Then there is an Heigenvector 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 Heigenvector 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 Heigenvector 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 Heigenvector 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 Heigenvalue.
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 Heigenvector 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 Heigenvector of