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 Jiang-Chao Wan School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China  and  Yi Wang School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China
###### 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:

 Axk =∑i1,i2,…,ik∈[n]ai1i2…ikxi1xi2⋯xik, (Axk−1)i =∑i2,…,ik∈[n]aii2…ikxi2⋯xik,i∈[n].

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

 ai1i2…ik={1(k−1)!,% if~{}{vi1,vi2,…,vik}∈E(G);0,otherwise.

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) Q(G)xk=∑e∈E(G)(xke+kxe),

and for each ,

 (Q(G)xk−1)v=d(v)xk−1v+∑e∈E(v)xe∖{v}.

So the eigenvector equation is equivalent to that for each ,

 (2.2) (λ−d(v))xk−1v=∑e∈E(v)xe∖{v}.

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

 (2.3) λmin(G)=minx∈Rn,∥x∥k=1∑e∈E(G)(xke+kxe).

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

 λmin(G)≤Q(G)(−x)k=−Q(G)xk=−ρ(Q(G))<0,

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,

 (λ−d(u))xku=∑e∈E(u)xe,(λ−d(v))xkv=∑e∈E(v)xe.

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

 ~xv=⎧⎪⎨⎪⎩xv,if v∈V(G0)∖{u};|xv|,if v∈U;−|xv|,if v∈W.

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

 λmin(G)≤Q(G)~xk

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

 ¯¯¯xv=⎧⎪⎨⎪⎩xv,if v∈V(G0)∖{u};−|xv|,if v∈U;|xv|,if v∈W.

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

 (λmin(G)−d(u))~xk−1u=0=∑e∈EG0(u)~xe∖{u}+∑e∈EH(u)~xe∖{u},
 (λmin(G)−d(u))¯xk−1u=0 = ∑e∈EG0(u)¯xe∖{u}+∑e∈EH(u)¯xe∖{u} = ∑e∈EG0(u)~xe∖{u}−∑e∈EH(u)~xe∖{u}.

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

 λmin(G0)≥λmin(G),

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

 ~yv={yv,if v∈V(G0);0,otherwise.
###### Proof.

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

 ¯yv=⎧⎨⎩yv,if v∈V(G0);yu,if v∈U∖{u};−yu,if v∈W.

Then , and

 Q(G)¯yk = ∑e∈E(G)(¯yke+k¯ye) = ∑e∈E(G0)(¯yke+k¯ye)+∑e∈E(H)(¯yke+k¯ye) = Q(G0)yk+∑e∈E(H)(kyku−kyku) = λmin(G0).

By Eq. (2.3), we have

 λmin(G)≤Q¯yk∥¯y∥kk=λmin(G0)1+(ν(H)−1)yku≤λmin(G0),

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

 λmin(G0)>λmin(G).
2. If is a first eigenvector of such that and , then

 λmin(G0)=λmin(G).
###### 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

 Q(G0)(x|G0)k−1=λmin(G)(x|G0)[k−1].

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) βH(x):=dH(u)xku+∑e∈EH(u)xe≤0.

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

###### Proof.

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

 (3.2) ((Q(G)−λI)xk−1)v=((Q(G0)−λI)(x|G0)k−1)v=0.

For the vertex ,

 λxk−1u=(Q(G)xk−1)u = dG(u)xk−1u+∑e∈EG(u)xe∖{u}, = dG0(u)xk−1u+∑e∈EG0(u)xe∖{u}+dH(u)xk−1u+∑e∈EH(u)xe∖{u} = (Q(G0)(x|G0)k−1)u+dH(u)xk−1u+∑e∈EH(u)xe∖{u}.

So,

 (3.3) ((Q(G0)−λI)(x|G0)k−1)u=−⎛⎝dH(u)xk−1u+∑e∈EH(u)xe∖{u}⎞⎠.

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

 0≤(Q(G0)−λI)(x|G0)k = (x|G0)⊤((Q(G0)−λI)(x|G0)k−1 = −xu⎛⎝dH(u)xk−1u+∑e∈EH(u)xe∖{u}⎞⎠ = −βH(x).

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,

 β¯T(x)=xku+xe=xku≤0,

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

 (1−λ)|xv|k=|xe|=|xu∥xv|k−2|xw|,

which implies that

 (3.4) |xv|=(|xu∥xw|1−λ)12.

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

 β¯T(x)=(dT(w)−1)xkw+∑~e∈ET(w)∖{e}x~e<0.

By Lemma 3.1(1), we have

 (3.5) (dT(w)−1)|xw|k<∑~e∈ET(w)∖{e}|x~e|.

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

 (dT(w)−λ)|xw|k = ∑~e∈ET(w)∖{e}|x~e|+|xu∥xv|k−2|xw| > (dT(w)−1)|xw|k+|xu|(|xu∥xw|1−λ)k2−1|xw| = (dT(w)−1)|xw|k+(1−λ)1−k2|xu|k2|xw|k2.

So

 (1−λ)|xw|k>(1−λ)1−k2|xu|k2|xw|k2,

and hence

 |xw|k2>(1−λ)k2|xw|k2>|xu|k2.

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) r=2m,2m−1,2m−2,…,2i,2i−1,2i−2,…,2,1,0,

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

 (3.7) |x2i|=fi(λmin(G))2k|x0|, i∈[m],

where is defined recursively as , ,

 fi+1(x)=(2−x)(1−x)k2−1fi(x)−fi−1(x), i∈[m−1].

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),

 |x0|=|x1|,|x2|=(1−λ)|x0|.

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

 (2−λ)|x2i|k−1 = |x2i−1|k−2|x2i−2|+|x2i+1|k−2|x2i+2| = (|x2ix2i−2|1−λ)k2−1|x2i−2|+(|x2ix2i+2|1−λ)k2−1|x2i+2| = (1−λ)1−k2|x2i|k2−1(|x2i−2|k2+|x2i+2|k2).

Thus, for ,

 |x2i+2|k2=(2−λ)(1−λ)k2−1|x2i|k2−|x2i−2|k2.

It is easy to verify that for ,

 |x2i|k2=fi(λ)|x0|k2.

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.

###### 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

 λmin(~G)≤λmin(G),

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