Spectral radius of uniform hypergraphs and degree sequences

Dongmei Chen; Zhibing Chen; Xiao-Dong Zhang

arXiv:1702.06350·math.CO·February 22, 2017

Spectral radius of uniform hypergraphs and degree sequences

Dongmei Chen, Zhibing Chen, Xiao-Dong Zhang

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TL;DR

This paper establishes new upper bounds for the spectral radii of uniform hypergraphs' adjacency and signless Laplacian matrices based on their degree sequences, advancing spectral hypergraph theory.

Contribution

It introduces novel bounds relating spectral radii to degree sequences in uniform hypergraphs, filling gaps in spectral hypergraph analysis.

Findings

01

Derived upper bounds for spectral radii based on degree sequences

02

Extended spectral bounds to both adjacency and signless Laplacian matrices

03

Enhanced understanding of spectral properties in uniform hypergraphs

Abstract

In this paper, we present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.

Equations56

c_{i α_{1} \dots α_{p - 1}} = i_{2}, \dots, i_{p} = 1 \sum n a_{i i_{2} \dots i_{p}} b_{i_{2} α_{1}} \dots b_{i_{p} α_{p - 1}}, i \in [n], α_{1}, \dots, α_{p - 1} \in [n]^{q - 1} .

c_{i α_{1} \dots α_{p - 1}} = i_{2}, \dots, i_{p} = 1 \sum n a_{i i_{2} \dots i_{p}} b_{i_{2} α_{1}} \dots b_{i_{p} α_{p - 1}}, i \in [n], α_{1}, \dots, α_{p - 1} \in [n]^{q - 1} .

T \cdot x = λ x^{[p]},

T \cdot x = λ x^{[p]},

a_{i_{1}\cdots i_{n}}=\left\{\begin{array}[]{ll}\frac{1}{(k-1)!},&\mbox{if}\ e=\{i_{1},\cdots,i_{k}\}\in E(\mathcal{H})\\ 0,&{\mbox{otherwise}.}\end{array}\right.

a_{i_{1}\cdots i_{n}}=\left\{\begin{array}[]{ll}\frac{1}{(k-1)!},&\mbox{if}\ e=\{i_{1},\cdots,i_{k}\}\in E(\mathcal{H})\\ 0,&{\mbox{otherwise}.}\end{array}\right.

min {R_{i} (T) : 1 \leq i \leq n} \leq ρ (T) \leq max {R_{i} (T) : 1 \leq i \leq n},

min {R_{i} (T) : 1 \leq i \leq n} \leq ρ (T) \leq max {R_{i} (T) : 1 \leq i \leq n},

\binom{n}{k}=\left\{\begin{array}[]{ll}\frac{n!}{k!(n-k)!},&\mbox{if $n\geq k\geq 0$}\\ 0&\mbox{ if $n<k$ or $k<0$.}\end{array}\right.

\binom{n}{k}=\left\{\begin{array}[]{ll}\frac{n!}{k!(n-k)!},&\mbox{if $n\geq k\geq 0$}\\ 0&\mbox{ if $n<k$ or $k<0$.}\end{array}\right.

r = 0 \sum k - 2 (k - r - 2 s - 3) (r n - s + 1) = p = 1 \sum k - 1 (p - 1 s - 2) (k - p - 1 n - p) = (k - 2 n - 2) .

r = 0 \sum k - 2 (k - r - 2 s - 3) (r n - s + 1) = p = 1 \sum k - 1 (p - 1 s - 2) (k - p - 1 n - p) = (k - 2 n - 2) .

r = 0 \sum k - 2 (r + 1) (r + 1 s - 1) (k - r - 2 n - s) = r = 0 \sum k - 2 (k - 1 - r) (k - r - 1 s - 2) (r n - s + 1) + (k - 2 n - 2) .

r = 0 \sum k - 2 (r + 1) (r + 1 s - 1) (k - r - 2 n - s) = r = 0 \sum k - 2 (k - 1 - r) (k - r - 1 s - 2) (r n - s + 1) + (k - 2 n - 2) .

r = 1 \sum n - 1 r (r n - 1) x^{r - 1} = r = 1 \sum s - 1 r (r s - 1) x^{r - 1} r = 0 \sum n - s (r n - s) x^{r} + r = 0 \sum s - 1 (r s - 1) x^{r} r = 1 \sum n - s r (r n - s) x^{r - 1} .

r = 1 \sum n - 1 r (r n - 1) x^{r - 1} = r = 1 \sum s - 1 r (r s - 1) x^{r - 1} r = 0 \sum n - s (r n - s) x^{r} + r = 0 \sum s - 1 (r s - 1) x^{r} r = 1 \sum n - s r (r n - s) x^{r - 1} .

(k - 1) (k - 1 n - 1) = r = 0 \sum k - 2 (r + 1) (r + 1 s - 1) (k - 2 - r n - s) + r = 0 \sum k - 2 (k - 1 - r) (r s - 1) (k - 1 - r n - s) .

(k - 1) (k - 1 n - 1) = r = 0 \sum k - 2 (r + 1) (r + 1 s - 1) (k - 2 - r n - s) + r = 0 \sum k - 2 (k - 1 - r) (r s - 1) (k - 1 - r n - s) .

[(1 + 1)^{n - 1}]^{'} = [(1 + x)^{s - 2}]^{'} (1 + x)^{n - s + 1} + (1 + x)^{s - 1} [(1 + x)^{n - s}]^{'} + (1 + x) (1 + x)^{n - 2} .

[(1 + 1)^{n - 1}]^{'} = [(1 + x)^{s - 2}]^{'} (1 + x)^{n - s + 1} + (1 + x)^{s - 1} [(1 + x)^{n - s}]^{'} + (1 + x) (1 + x)^{n - 2} .

(k - 1) (k - 1 n - 1)

(k - 1) (k - 1 n - 1)

ρ (H) \leq 1 \leq s \leq n min {ϕ_{s} = \frac{d _{s} - A _{2} + ( s - 2 ) A _{1} + Δ}{2}} .

ρ (H) \leq 1 \leq s \leq n min {ϕ_{s} = \frac{d _{s} - A _{2} + ( s - 2 ) A _{1} + Δ}{2}} .

R_{j} (B)

R_{j} (B)

\leq

\leq

R_{j} (B)

R_{j} (B)

\leq

\leq

ϕ_{s}^{2} - [d_{s} - A_{2} + (s - 2) A_{1}] ϕ_{s} - A_{1} [(d_{s} - A_{2} + (s - 1) A_{1} + t = 1 \sum s - 1 (d_{t} - d_{s})] = 0.

ϕ_{s}^{2} - [d_{s} - A_{2} + (s - 2) A_{1}] ϕ_{s} - A_{1} [(d_{s} - A_{2} + (s - 1) A_{1} + t = 1 \sum s - 1 (d_{t} - d_{s})] = 0.

t = 1 \sum s - 1 x_{t}^{k - 1} = \frac{ϕ _{s} - d _{s} + A _{2}}{A _{1}}

t = 1 \sum s - 1 x_{t}^{k - 1} = \frac{ϕ _{s} - d _{s} + A _{2}}{A _{1}}

R_{j} (B)

R_{j} (B)

R_{1} (B) \leq \frac{d _{1}}{x _{1}^{k - 1}}

R_{1} (B) \leq \frac{d _{1}}{x _{1}^{k - 1}}

R_{j} (H) \leq d_{2} + A_{1} x_{1}^{k - 1} - A_{1}, \mbox f or j = 2, \dots, n .

R_{j} (H) \leq d_{2} + A_{1} x_{1}^{k - 1} - A_{1}, \mbox f or j = 2, \dots, n .

R_{1} (B) \leq \frac{d _{1}}{x _{1}^{k - 1}} = ϕ_{2}

R_{1} (B) \leq \frac{d _{1}}{x _{1}^{k - 1}} = ϕ_{2}

R_{j} (H) \leq d_{2} + A_{1} x_{1}^{k - 1} - A_{1} = ϕ_{2}, \mbox f or j = 2, \dots, n .

R_{j} (H) \leq d_{2} + A_{1} x_{1}^{k - 1} - A_{1} = ϕ_{2}, \mbox f or j = 2, \dots, n .

ρ (H) \leq \frac{δ + \frac{n - 2}{k - 1} ( k - 2 n - 2 ) - ( k - 2 n - 1 ) + ( δ + \frac{n}{k - 1} ( k - 2 n - 2 ) - ( k - 2 n - 1 ) ) ^{2} + \frac{4}{k - 1} ( k - 2 n - 2 ) ( k m - n δ )}{2} .

ρ (H) \leq \frac{δ + \frac{n - 2}{k - 1} ( k - 2 n - 2 ) - ( k - 2 n - 1 ) + ( δ + \frac{n}{k - 1} ( k - 2 n - 2 ) - ( k - 2 n - 1 ) ) ^{2} + \frac{4}{k - 1} ( k - 2 n - 2 ) ( k m - n δ )}{2} .

q (H) \leq 1 \leq s \leq n min {ψ_{s} = \frac{2 d _{s} - A _{2} + ( s - 2 ) A _{1} + Θ}{2}} .

q (H) \leq 1 \leq s \leq n min {ψ_{s} = \frac{2 d _{s} - A _{2} + ( s - 2 ) A _{1} + Θ}{2}} .

R_{j} (M) \leq ψ_{2} \mbox f or j = 1, \dots, n .

R_{j} (M) \leq ψ_{2} \mbox f or j = 1, \dots, n .

R_{j} (C) \leq ψ_{s} \mbox f or j = 1, \dots, n .

R_{j} (C) \leq ψ_{s} \mbox f or j = 1, \dots, n .

q (H) \leq \frac{2 δ + \frac{n - 2}{k - 1} ( k - 2 n - 2 ) - ( k - 2 n - 1 ) + ( 2 δ + \frac{n}{k - 1} ( k - 2 n - 2 ) - ( k - 2 n - 1 ) ) ^{2} + \frac{8}{k - 1} ( k - 2 n - 2 ) ( k m - n δ )}{2} .

q (H) \leq \frac{2 δ + \frac{n - 2}{k - 1} ( k - 2 n - 2 ) - ( k - 2 n - 1 ) + ( 2 δ + \frac{n}{k - 1} ( k - 2 n - 2 ) - ( k - 2 n - 1 ) ) ^{2} + \frac{8}{k - 1} ( k - 2 n - 2 ) ( k m - n δ )}{2} .

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Taxonomy

TopicsTensor decomposition and applications · Graph theory and applications · Matrix Theory and Algorithms

Full text

Spectral Radius of Uniform Hypergraphs and Degree Sequences

††thanks: This work is supported by the National Natural Science Foundation of China (Nos.11531001 and 11271256), the Joint NSFC-ISF Research Program (jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (No. 11561141001)), Innovation Program of Shanghai Municipal Education Commission (No. 14ZZ016) and Specialized Research Fund for the Doctoral Program of Higher Education (No.20130073110075).

Dong-Mei Chen1, Zhi-Bing Chen1, Xiao-Dong Zhang2

1College of Mathematics and Statistics

Shenzhen University

3688 Nanhai Road, Shenzhen 518060, P.R. China

2School fo Mathematical Sciences, MOE-LSC, SHL-MAC

Shanghai Jiao Tong University

800 Dongchuan road, Shanghai, 200240, P.R. China Corresponding author (E-mail address: [email protected])

Abstract

In this paper, we present upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.

Key words: Spectral radius; Uniform hypergraph; Degree sequence.

AMS Classifications: 05C50, 05C05, 05C40.

1 Introduction

Let ${\mathcal{H}}=(V(\mathcal{H}),E(\mathcal{H}))$ be a simple (i.e., no loops or multiedges) hypergraph, where the vertex set $V(\mathcal{H})=[n]:=\{1,\cdots,n\}$ and the edge set $E(\mathcal{H})=\{e_{1},\cdots,e_{m}\}$ with $e_{i}\subseteq V(\mathcal{H})$ for $i=1,\cdots,m$ . Further, if $|e_{i}|=k$ for $i=1,\cdots,m$ , then $\mathcal{H}$ is called a $k-$ uniform hypergraph. The degree of vertex $v\in V(\mathcal{H})$ in a hypergraph $\mathcal{H}$ , written $d_{v}$ , is the number of edges incident to $v$ , i.e., $d_{v}=|\{e\in E(\mathcal{H})\ \ |\ \ v\in e\}|$ . The sequence $(d_{1},\cdots,d_{n})$ is called degree sequence of $\mathcal{H}$ . If the degree of each vertex is equal to $d$ , i.e., $d_{1}=\cdots=d_{n}=k$ , then $\mathcal{H}$ is called a $d-$ reguar hypergraph.

Let $n$ and $p$ be two positive integers. An order $p$ and dimension $n$ tensor $\mathcal{A}=(a_{i_{1}i_{2}\cdots i_{p}})$ over the complex field $C$ is a multidimensional array with all entries $a_{i_{1}i_{2}\cdots i_{p}}\in C$ for $i_{j}=1,\cdots,n$ and $j=1,\cdots,p$ . Clearly, a tensor of order $1$ and dimension $n$ is a vector and a tensor of order $2$ and dimension $n$ is an $n\times n$ matrix. For the sake of simplicity, $a_{i_{1}i_{2}\cdots i_{p}}$ is denoted by $a_{i_{1}\alpha},$ where $\alpha=i_{2}\cdots i_{p}\in[n]^{p-1}$ . If $\mathcal{A}$ and $\mathcal{B}$ are two tensors of order $p\geq 2$ and order $q\geq 1$ , dimension $n$ , respectively, the product of two tensors $\mathcal{A}$ and $\mathcal{B}$ (see [13]) is defined to be the tensor $\mathcal{C}=(c_{i\alpha_{1}\cdots\alpha_{m-1}}):=\mathcal{A}\cdot\mathcal{B}$ is an order $(p-1)(q-1)$ and dimension $n$ , where

[TABLE]

For a tensor $\mathcal{T}$ of order $p\geq 2$ and dimension $n$ , if there exists a complex number $\lambda$ and a vector $x=(x_{1},\cdots,x_{n})^{T}$ (i.e. a tensor of order $1$ and dimension $n$ such that

[TABLE]

where $x^{[p]}=(x^{p-1},\cdots,x^{p-1})^{T}$ , then $\lambda$ is called an eigenvalue of $\mathcal{T}$ and $x$ is called an eigenvector of $\mathcal{T}$ corresponding to the eigenvalue $\lambda$ (for example, see [10]). The largest modulus of eigenvalues of $\mathcal{T}$ is called spectral radius of $\mathcal{T}$ and denoted by $\rho(\mathcal{T})$ . It is known (for example, see [15]) that for a nonnegative tensor $\mathcal{T}$ , $\rho(\mathcal{T})$ is a nonnegative eigenvalue and corresponding to a nonnegative eigenvector. The readers may refer to an excellent survey [1] for spectral theory of nonnegative tensors.

For a hypergraph $\mathcal{H}$ , there are a few tensors associated with $\mathcal{H}$ . The most important tensor associated with $\mathcal{H}$ may be the adjacency tensor. The adjacency tensor of a $k-$ uniform hypergraph $\mathcal{H}$ on $n$ vertices is defined as the tensor $\mathcal{A}(\mathcal{H})=(a_{i_{1}\cdots i_{k}})$ of order $k$ and dimension $n$ , where

[TABLE]

The spectral radius of the adjacency matrix $\mathcal{A}(\mathcal{H})$ of a $k-$ uniform hypergraph is called spectral radius of $\mathcal{H}$ and denoted by $\rho(\mathcal{H})$ . The spectral theory of hypergraph has received more and more attention. For example, Cooper and Dutle [2] gave an excellent survey on the spectral theory of $k-$ uniform hypergraph. Another important tensor associated with $k-$ uniform hypergraph $\mathcal{H}$ is signless Laplacian tensor. Let $\mathcal{D}(\mathcal{H})$ be a $k-$ th order $n-$ dimensional diagonal tensor whose diagonal entry $d_{1},\cdots,d_{n}$ . Then $\mathcal{Q}(\mathcal{H})=\mathcal{D}(\mathcal{H})+\mathcal{A}(\mathcal{H})$ is called the signless Laplacian tensor of $H$ . The spectral radius of $\mathcal{Q}(\mathcal{H})$ is called the signless Lapalcian spectral radius of $\mathcal{H}$ and denoted by $q(\mathcal{H})$ . Li et. al. [8] gave some upper bounds for the $H-$ spectral radius and $Z-$ spectral radius of uniform hypergraphs in terms of parameters such as number of vertices, number of edges, maximum degree, and minimum degree. Yuan et.al. [16] presented upper bounds for spectral radius and signless spectral raidus of hypergraphs in terms of the degrees of vertices. The related results may be referred to [3, 4, 5, 7, 9, 17].

In this paper, we present some upper bound for spectral radius and signless Laplacian spectral radius of a $k-$ unform hypergraph in terms of degree sequences, which extends some known results on hypergraphs. In Section 2, Some preliminaries and Lemmas are presented. In Section 3, we present the main results of this paper and proof.

2 Preliminaries

In this section, we present some known results and lemmas

Lemma 2.1

[15, 4]** Let $\mathcal{T}=(a_{i_{1}\cdots i_{k}})$ be an order $k$ and dimension $n$ tensor with $k\geq 2$ . Then

[TABLE]

where $R_{i}(\mathcal{T})$ is the sum of row $i$ in $\mathcal{T}$ , i.e., $R_{i}(\mathcal{T})=\sum_{i_{2},\cdots,i_{k}=1}^{n}a_{ii_{2}\cdots i_{k}}$ . Moreover, if $\mathcal{T}$ is weakly irreducible, then either equality holds if and only if $R_{1}(\mathcal{T})=\cdots=R_{n}(\mathcal{T})$ .

Lemma 2.2

([13]) Let $\mathcal{A}$ and $\mathcal{B}$ be two order $k$ dimension $n$ tensors. If there exists a nonsingular diagonal matrix $D$ such that $\mathcal{B}=D^{-(k-1)}\cdot\mathcal{A}\cdot D$ , then $\mathcal{A}$ and $\mathcal{B}$ have the same eigenvalues including multiplicity. In particular, they have the same spectral radius.

For convenience, if $n$ and $k$ are two integers,

[TABLE]

The following identity equality is known.

Lemma 2.3

([14]) Let $n>s\geq 3$ and $kge2$ be three positive integers. Then

[TABLE]

Lemma 2.4

If $n,k,s$ are three positive integers with $n\geq s\geq 3$ , then

[TABLE]

Proof. Since $(1+x)^{n-1}=(1+x)^{s-1}(1+x)^{n-s},$ we differentiate both side with respect to $x$ and have

[TABLE]

Hence considering the coefficients of $x^{k-2}$ of both side of equation (4), we have

[TABLE]

On the other hand, Since $(1+x)^{n-1}=(1+x)^{s-2}(1+x)(1+x)^{n-s},$ we differentiate both side with respect to $x$ and have

[TABLE]

Then considering the coefficients of $x^{k-2}$ both side of equation (6), we have

[TABLE]

By (5) and (7), it is easy to see that (4) holds.

3 Main Results

In this section, we present the main result of this paper as follows.

Theorem 3.1

Let $\mathcal{H}$ be a $k-$ uniform hypergraph with degree sequence $d_{1}\geq d_{2}\geq\cdots\geq d_{n}$ . Denote $A_{1}=\frac{1}{k-1}\binom{n-2}{k-2}$ , $A_{2}=\sum_{r=0}^{k-2}\frac{r+1}{k-1}\binom{s-1}{r+1}\binom{n-s}{k-r-2}$ , $\Delta=(d_{s}+sA_{1}-A_{2})^{2}+4A_{1}\sum_{t=1}^{s-1}(d_{t}-d_{s})$ and $\phi_{s}=\frac{d_{s}-A_{2}+(s-2)A_{1}+\sqrt{\Delta}}{2}$ , for $1\leq s\leq n$ . Then

[TABLE]

Proof. We first consider $s\geq 3$ . Let $\mathcal{A}$ be the adjacency matrix of $\mathcal{H}$ and $D=diag(x_{1},\cdots,x_{n})$ be the diagonal matrix with $x_{s}=\cdots=x_{n}=1$ . Let $\mathcal{B}=D^{-(k-1)}\cdot\mathcal{A}\cdot D=(b_{i_{1}\cdots i_{k}})$ . By Lemma 2.2, $\rho(\mathcal{H})=\rho(\mathcal{A})=\rho(\mathcal{B})$ . For $0\leq r\leq n$ , denote $I_{r}=\{\{i_{2},\cdots,i_{k}\}\ |\ 1\leq i_{2},\cdots,i_{k}\leq n,\mbox{ there are exactly }\ r\ \mbox{elements in }\ \{i_{2},\cdots,i_{k}\}$ with at least $\ s\}$ . The sum of row $j$ in the tensor $\mathcal{B}$ is denoted by $R_{j}(\mathcal{B})$ for $1\leq j\leq n$ . We consider the following two cases

Case 1: $1\leq j\leq s-1$ . By (2) in Lemma 2.3, $x_{s}=\cdots=x_{n}=1$ , we have

[TABLE]

Case 2: $s\leq j\leq n$ . By (2) in Lemma 2.3, $x_{s}=\cdots=x_{n}=1$ , we have

[TABLE]

Let $x_{t}^{k-1}=1+\frac{d_{t}-d_{s}}{A_{1}+\phi_{s}}$ . By the definition of $\phi_{s}$ , it is easy to see that

[TABLE]

Hence $d_{s}+A_{1}\sum_{t=1}^{s-1}x_{t}^{k-1}-A_{2}=\phi_{s}$ , which implies that $R_{j}(\mathcal{B})\leq\phi_{s}$ for $s\leq j\leq n$ . Moreover, for $1\leq j\leq s-1$ , we have

[TABLE]

and

[TABLE]

Hence by Lemma 2.1, we have $\rho(\mathcal{B})\leq\phi_{s}$ for $3\leq s\leq n$ .

If $s=1$ , then it is easy to see that $\phi_{1}=d_{1}$ . By [2], we have $\rho(\mathcal{H})\leq d_{1}=\phi_{1}$ .

If $s=2$ , then by the same argument as $s\geq 3$ , it is easy to see that

[TABLE]

and

[TABLE]

Let $x_{1}^{k-1}=\frac{-(d_{2}-A_{1})+\sqrt{(d_{2}-A_{1})^{2}+4A_{1}d_{1}}}{2A_{1}}$ . Then

[TABLE]

and

[TABLE]

Hence $\rho(\mathcal{H})\leq\phi_{2}$ . Therefore, $\rho(\mathcal{H})\leq\phi_{s}$ for $s=1,\cdots,n$ .

Remark The The sequence $\phi_{1},\cdots,\phi_{n}$ is not necessarily non-increasing. In particular, we are able to get an upper bound in terms of then minimum degree $\delta$ and the size of edges.

Corollary 3.2

Let $\mathcal{H}$ be a $k-$ uniform hypergraph with the minimum degree $\delta$ and the size $m$ of edges. Then

[TABLE]

Proof. The assertion follows from $\rho(H)\leq\phi_{n}$ in Theorem 3.1.

Similarly, we are able to get an upper bound for the signless Laplacian spectral radius of $Q(\mathcal{H})$ .

Theorem 3.3

Let $\mathcal{H}$ be a $k-$ uniform hypergraph with degree sequence $d_{1}\geq d_{2}\geq\cdots\geq d_{n}$ and $q(\mathcal{H})$ be the spectral radius of $Q(\mathcal{H})$ . Denote $A_{1}=\frac{1}{k-1}\binom{n-2}{k-2}$ , $A_{2}=\sum_{r=0}^{k-2}\frac{r+1}{k-1}\binom{s-1}{r+1}\binom{n-s}{k-r-2}$ , $\Theta=(2d_{s}+sA_{1}-A_{2})^{2}+8A_{1}\sum_{t=1}^{s-1}(d_{t}-d_{s})$ and $\psi_{s}=\frac{2d_{s}-A_{2}+(s-2)A_{1}+\sqrt{\Theta}}{2}$ , for $1\leq s\leq n$ . Then

[TABLE]

Proof. Clearly, $\psi_{1}=2d_{1}$ . Hence $q(\mathcal{H})\leq\psi_{1}$ by [18]. Let $Y=diag(y_{1},1,\cdots,1)$ be the diagonal matrix with $y_{1}^{k-1}=\frac{-(2d_{2}-A_{1})+\sqrt{(2d_{2}-A_{1})^{2}+8A_{1}d_{1}}}{2A_{1}}$ and $\mathcal{M}=Y^{k-1}\cdot q(\mathcal{H})\cdot Y$ . Then by the same argument in Theorem 3.1, it is easy to see that

[TABLE]

Hence $q(\mathcal{H})\leq\psi_{2}.$

We now consider $s\geq 3$ . Let $X=diag(x_{1},\cdots,x_{n})$ be the diagonal matrix with $x_{j}=1+\frac{2(d_{j}-d_{s})}{A_{1}+\psi_{s}}$ for $j=1,\cdots,s-1$ and $x_{s}=\cdots=x_{n}=1$ . Let $\mathcal{C}=X^{-(k-1)}\cdot Q(\mathcal{H})\cdot X$ . For $1\leq j\leq s-1$ , by the same argument in Theorem 3.1, we have

[TABLE]

Hence $q(\mathcal{H})\leq\psi_{s}$ . So the assertion holds.

Corollary 3.4

Let $\mathcal{H}$ be a $k-$ uniform hypergraph with the minimum degree $\delta$ and the size $m$ of edges. Then

[TABLE]

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