Least H-eigenvalue of hypergraphs

The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs

Yi-Zheng Fan School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China fanyz@ahu.edu.cn Jiang-Chao Wan School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China 1500256209@qq.com  and  Yi Wang School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China wangy@ahu.edu.cn
Abstract.

Let be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of is simply called the least eigenvalue of and the corresponding H-eigenvectors are called the first eigenvectors of . In this paper we give some numerical and structural properties about the first eigenvectors of which contains an odd-bipartite branch, and investigate how the least eigenvalue of changes when an odd-bipartite branch attached at one vertex is relocated to another vertex. We characterize the hypergraph(s) whose least eigenvalue attains the minimum among a certain class of hypergraphs which contain a fixed non-odd-bipartite connected hypergraph. Finally we present some upper bounds of the least eigenvalue and prove that zero is the least limit point of the least eigenvalues of connected non-odd-bipartite hypergraphs.

Key words and phrases:
Hypergraph, signless Laplacian tensor, least H-eigenvalue, eigenvector, odd-bipartite, perturbation
2000 Mathematics Subject Classification:
Primary 15A18, 05C65; Secondary 13P15, 05C15
2010 Mathematics Subject Classification:
Primary 15A18, 05C65; Secondary 13P15, 14M99
The corresponding author. This work was supported by National Natural Science Foundation of China (Grant No. 11871073, 11771016).

1. Introduction

Since Lim [13] and Qi [16] independently introduced the eigenvalues of tensors or hypermatrices in 2005, the spectral theory of tensors developed rapidly, especially the well-known Perron-Frobenius theorem of nonnegative matrices was generalized to nonnegative tensors [2, 6, 20, 21, 22]. The signless Laplacian tensors [17] were introduced to investigating the structure of hypergraphs, just like signless Laplacian matrices to simple graphs. As is nonnegative, by using Perron-Frobenius theorem, many results about its spectral radius are presented [9, 10, 12, 14, 23].

Let be a -uniform connected hypergraph. Shao et al. [18] prove that zero is an H-eigenvalue of if and only if is even and is odd-bipartite. Some other equivalent conditions are summarized in [5]. Note that zero is an eigenvalue of if and only if is even and is odd-colorable [5]. So, there exist odd-colorable but non-odd-bipartite hypergraphs [4, 15], for which zero is an N-eigenvalue. Hu and Qi [7] discuss the H-eigenvectors of zero eigenvalue of related to the odd-bipartitions of , and use N-eigenvectors of zero eigenvalue of to discuss some kinds of partition of , where an eigenvector is called H-(or N-)eigenvector if it can (or cannot) be scaled into a real vector.

Except the above work, the least H-eigenvalue of receives little attention. In this paper, we focus on the least H-eigenvalue of . Qi [16] proved that each eigenvalue of of a connected -uniform hypergraph has a nonnegative real part by using Gershgorin disks, which implies that the least H-eigenvalue of is at least zero, and is zero if and only if is even and is odd-bipartite. If is even, then are positive semi-definite [17], and its least H-eigenvalue is a solution of minimum problem over a real unit sphere; see Eq. (2.3). So, throughout of this paper, when discussing the least H-eigenvalue of , we always assume that is connected non-odd-bipartite with even uniformity . For convenience, the least H-eigenvalue of is simply called the least eigenvalue of and the corresponding H-eigenvectors are called the first eigenvectors of .

In this paper we give some numerical and structural properties about the first eigenvectors of which contains an odd-bipartite branch, and investigate how the least eigenvalue of changes when an odd-bipartite branch attached at one vertex is relocated to another vertex. We characterize the hypergraph(s) whose least eigenvalue attains the minimum among a certain class of hypergraphs which contain a fixed non-odd-bipartite connected hypergraph. Finally we present some upper bounds of the least eigenvalue and prove that zero is the least limit point of the least eigenvalues of connected non-odd-bipartite hypergraphs. The perturbation result on the least eigenvalue in this paper is a generalization of that on the least eigenvalue of the signless Laplacian matrix of a simple graph in [19].

2. Preliminaries

2.1. Eigenvalues of tensors

A real tensor (also called hypermatrix) of order and dimension refers to a multi-dimensional array with entries for all and . Obviously, if , then is a square matrix of dimension . The tensor is called symmetric if its entries are invariant under any permutation of their indices.

Given a vector , and , which are defined as follows:

Let be the identity tensor of order and dimension , that is, if and otherwise.

Definition 2.1 ([13, 16]).

Let be a real tensor of order dimension . For some , if the polynomial system , or equivalently , has a solution , then is called an eigenvalue of and is an eigenvector of associated with , where .

In the above definition, is called an eigenpair of . If is a real eigenvector of , surely the corresponding eigenvalue is real. In this case, is called an H-eigenvalue of . Denote by the least H-eigenvalue of .

A real tensor of even order is called positive semidefinite (or positive definite) if for any , (or ).

Lemma 2.2 ([16], Theorem 5).

Let be a real symmetric tensor of order and dimension , where is even. Then the following results hold.

  1. always has H-eigenvalues, and is positive definite (or positive semidefinite) if and only if its least H-eigenvalue is positive (or nonnegative).

  2. , where . Furthermore, is an optimal solution of the above optimization if and only if it is an eigenvector of associated with .

2.2. Uniform hypergraphs

A hypergraph is a pair consisting of a vertex set and an edge set , where for each . If for all , then is called a -uniform hypergraph. The degree or simply of a vertex is defined as . The order of is the cardinality of , denoted by , and its size is the cardinality of , denoted by . A walk in a is a sequence of alternate vertices and edges: , where for . A walk is called a path if all the vertices and edges appeared on the walk are distinct. A hypergraph is called connected if any two vertices of are connected by a walk or path.

If a hypergraph is both connected and acyclic, it is called a hypertree. The -th power of a simple graph , denoted by , is obtained from by replacing each edge (a -set) with a -set by adding additional vertices [8]. The -th power of a tree is called power hypertree, which is surely a -uniform hypertree. In particular, the -th power of a path (respectively, a star ) (as a simple graph) with edges is called a hyperpath (respectively, hyperstar), denote by (respectively, ). In a -th power hypertree , an edge is called a pendent edge of if it contains vertices of degree one, which are called the pendent vertices of .

Lemma 2.3 ([1]).

Let be a connected -uniform hypergraph. Then is a hypertree if and only if

The odd-bipartite hypergraphs was introduced by Hu and Qi [7], which is considered as a generalization of the ordinary bipartite graphs. The odd-bipartition is closely related to odd-traversal [15].

Definition 2.4 ([7]).

Let be even. A -uniform hypergraph is called odd-bipartite, if there exists a bipartition of such that each edge of intersects (or ) in an odd number of vertices (such bipartition is called an odd-bipartition of ); otherwise, is called non-odd-bipartite.

Let be a -uniform hypergraph on vertices . The adjacency tensor of [3] is defined as , an order dimensional tensor, where

Let be a diagonal tensor of order and dimension , where for . The tensor is called the signless Laplacian tensor of [17]. Observe that the adjacency (signless Laplacian) tensor of a hypergraph is symmetric.

Let . Then can be considered as a function defined on the vertices of , that is, each vertex is mapped to . If is an eigenvector of , then it defines on naturally, i.e., is the entry of corresponding to . If is a sub-hypergraph of , denote by the restriction of on the vertices of , or a subvector of indexed by the vertices of .

Denote by , or simply , the set of edges of containing . For a subset of , denote , and . Then we have

(2.1)

and for each ,

So the eigenvector equation is equivalent to that for each ,

(2.2)

From Lemma 2.2(2), if is even, then can be expressed as

(2.3)

Note that if is odd, the Eq. (2.3) does not hold. The reason is as follows. If contains at least one edge, then by Perron-Frobenius theorem, the spectral radius of is positive associated with a unit nonnegative eigenvector . Now

a contradiction as (see [17, Theorem 3.1]).

Lemma 2.5.

Let be a -uniform hypergraph, and be an eigenpair of . If and , then .

Proof.

Consider the eigenvector equation of at and respectively,

As , and . The result follows. ∎

Lemma 2.6 ([11]).

Let be a -uniform hypergraph with the minimum degree , where is even. Then .

3. Properties of the first eigenvectors

Let , be two vertex-disjoint hypergraphs, and let . The coalescence of , with respect to , denoted by , is obtained from , by identifying with and forming a new vertex . The graph is also written as . If a connected graph can be expressed in the form , where , are both nontrivial and connected, then is called a branch of with root . Clearly is also a branch of with root in the above definition.

We will give some properties of the first eigenvectors of a connected -uniform which contains an odd-bipartite branch. We stress that is even in this and the following sections.

Lemma 3.1.

Let be a connected -uniform hypergraph, where is odd-bipartite. Let be a first eigenvector of . Then the following results hold.

  1. for each .

  2. If , then , and for each .

  3. There exists a first eigenvector of ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡such that it is nonnegative on one part and nonpositive on the other part for any odd-bipartition of .

Proof.

Let be an odd-bipartition of , where . Without loss of generality, we assume that and . Let be such that

Note that , and for each ,

  • .

  • with equality if and only if .

We prove the assertion (1) by a contradiction. Suppose that there exists an edge such that . Then . By (a), (b), and Eq. (2.3), we have

a contradiction. So for each , and is also a first eigenvector as . The assertions (1) and (3) follow.

For the assertion (2), let be such that

By a similar discussion, is also a first eigenvector of . Note that and consider the eigenvector equation Eq. (2.2) of and at , respectively.

Thus and . As for each edge , we have for each . The assertion (2) follows by the definition of . ∎

Lemma 3.2.

Let be a connected non-odd-bipartite -uniform hypergraph, where is odd-bipartite. Then

with equality if and only if for any first eigenvector of , and is a first eigenvector of , where is defined by

Proof.

Suppose that is a first eigenvector of , , and . Let be an odd-bipartition of , where . Define by

Then , and

By Eq. (2.3), we have

where the first equality holds if and only if is also a first eigenvector of , and the second equality holds if and only if (Note that as is connected and non-odd-bipartite). The result now follows. ∎

Corollary 3.3.

Let be a connected non-odd-bipartite -uniform hypergraph, where is odd-bipartite.

  1. If is a first eigenvector of with , then

  2. If is a first eigenvector of such that and , then

Proof.

By Lemma 3.2, we can get the assertion (1) immediately. Let be a first eigenvector of as in (2). By Lemma 3.1(2), . Considering the eigenvector equation (2.2) of at each vertex of , we have

So is an eigenvector of associated with the eigenvalue . The result follows by Lemma 3.2. ∎

Lemma 3.4.

Let be a connected non-odd-bipartite -uniform hypergraph, where is odd-bipartite. If is a first eigenvector of , then

(3.1)

Furthermore, if and , then and ; or equivalently if , then .

Proof.

Let . By Eq. (2.2), for each ,

(3.2)

For the vertex ,

So,

(3.3)

By Lemma 3.2 and Lemma 2.2(1), is positive semidefinite. Then for any real and nonzero . So, by Eq. (3.2) and Eq. (3.3), we have

So we have .

Suppose that and . If , by Corollary 3.3(2), . If , then . By Eq. (3.2) and Eq. (3.3), is an eigenpair of , implying that by Lemma 3.2. However, , a contradiction to Corollary 3.3(1). ∎

Lemma 3.5.

Let be a connected non-odd-bipartite -uniform hypergraph, where is a power hypertree. If is a first eigenvector of and for some , then whenever is a vertex of such that lies on the unique path from to .

Proof.

It suffices to consider three vertices in a common edge , where , , , and lies on the path from to . We will show . Write , where contains as a sub-hypergraph, and is a sub-hypergraph of such that is the only edge of containing . Suppose that . If , by Lemma 3.4,

a contradiction. So . If , then by Eq. (2.2), as by Lemma 2.6, also a contradiction. So . ∎

Lemma 3.6.

Let be a connected non-odd-bipartite -uniform hypergraph, where is a power hypertree. If is a first eigenvector of and , then whenever are two vertices of such that lies on the unique path from to , and .

Proof.

By Lemma 3.5, for any vertex . It suffices to consider three vertices in a common edge , where , , , and lies on the path from to . We will show that . By the eigenvector equation of at , noting that , by Lemma 2.5 we have

which implies that

(3.4)

We can write , where contains and the edge as the only one containing . Then by Lemma 3.4, noting ,

By Lemma 3.1(1), we have

(3.5)

Considering the eigenvector equation of at , by Lemma 2.5, Eq. (3.4) and Eq. (3.5), we have

So

and hence

Denote by the coalescence of and by identifying one vertex of and one pendent vertex of and forming a new vertex .

Lemma 3.7.

Let be a connected non-odd-bipartite -uniform hypergraph, where is a hyperpath with edges. Starting from the root , label edges of as , and some vertices of those edges as

(3.6)

where for , and for , . If is a first eigenvector of and , Then

(3.7)

where is defined recursively as , ,

Furthermore, , and is strictly decreasing in .

Proof.

By Lemma 3.5, for each . Let . Then by Lemma 2.6. By Lemma 2.5 and Eq. (2.2),

So, by Lemma 2.5, Lemma 3.1(1) and Eq. (3.4), considering the eigenvector equation of at the vertex (), we have

Thus, for ,

It is easy to verify that for ,

By Lemma 3.6, for , , and is strictly decreasing in . ∎

4. Perturbation of the least eigenvalues

We first give a perturbation result on the least eigenvalues under relocating an odd-bipartite branch. Let , be two vertex-disjoint hypergraphs, where , are two distinct vertices of , and is a vertex of (called the root of ). Let and . We say that is obtained from by relocating rooted at from to ; see Fig. 4.1.

Figure 4.1. Relocating from to
Lemma 4.1.

Let and be connected non-odd-bipartite -uniform hypergraphs, where is odd-bipartite. If is a first eigenvector of such that , then

with equality if and only if , and defined in (4.4) is a first eigenvector of .

Proof.

Let be a first eigenvector of such that and . We divide the discussion into three cases. Denote .

Case 1: . Write