Principal eigenvectors and spectral radii of uniform hypergraphs

Linear Algebra and its Applications 544 (2018) 273–285

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Principal eigenvectors and spectral radii of uniform hypergraphs Haifeng Li a , Jiang Zhou b , Changjiang Bu a,b,∗ a b

College of Automation, Harbin Engineering University, Harbin 150001, PR China College of Science, Harbin Engineering University, Harbin 150001, PR China

a r t i c l e

i n f o

Article history: Received 1 August 2016 Accepted 11 January 2018 Submitted by R. Brualdi MSC: 05C50 05C65 15A69 15A18

a b s t r a c t In this paper, some inequalities among the principal eigenvector, spectral radius and vertex degrees of a connected uniform hypergraph are established. Necessary and sufficient conditions of equalities holding are presented, which are related to the regularity of a hypergraph. Furthermore, we present some bounds on the spectral radius for a connected irregular uniform hypergraph in terms of some parameters, such as principal ratio, maximum degree, diameter, and the number of vertices and edges. © 2018 Elsevier Inc. All rights reserved.

Keywords: Hypergraph Spectral radius Principal eigenvector

1. Introduction For a positive integer n, let [n] = {1, 2, . . . , n}. An order m dimension n tensor A = (ai1 i2 ···im ) is a multidimensional array with nm entries, where ij ∈ [n], j ∈ [m]. When m = 2, A is an n × n matrix. Let C[m,n] be the set of order m dimension n tensors * Corresponding author. E-mail address: [email protected] (C. Bu). https://doi.org/10.1016/j.laa.2018.01.017 0024-3795/© 2018 Elsevier Inc. All rights reserved.

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274

over the complex field C, and Cn be the set of n-vectors over the complex field C. For A = (ai1 i2 ···im ) ∈ C[m,n] , if all the entries ai1 i2 ···im ≥ 0, then A is called nonnegative. In 2005, Qi [22] and Lim [14] defined the eigenvalues of tensors, respectively. For T A = (ai1 i2 ···im ) ∈ C[m,n] and x = (x1 , x2 , . . . , xn ) ∈ Cn , Axm−1 is an n-vector whose the i-th component is  m−1  Ax = i

n 

aii2 ···im xi2 · · · xim .

i2 ,...,im =1 T

If there exists a number λ ∈ C and a nonzero vector x = (x1 , . . . , xn ) ∈ Cn such that Axm−1 = λx[m−1] , then λ is called an eigenvalue of A, x is called an eigenvector of A corresponding to λ,  T where x[m−1] = xm−1 , xm−1 , . . . , xm−1 . The spectral radius ρ(A) = max{|λ| : λ ∈ n 1 2 σ(A)}, where σ (A) is the set of all eigenvalues of A. A hypergraph G is a pair (V (G), E(G)), where E(G) ⊆ P (V (G)) and P (V (G)) stands for the power set of V (G). The elements of V (G) are called the vertices and the elements of E(G) are called the edges (see [1]). If each edge of G contains exactly k distinct vertices, then G is called k-uniform. When k = 2, G is a graph. For all i ∈ V (G), Ei (G) denotes the set of edges containing i, and di = |Ei (G)| denotes the degree of i, Δ = max{di } and i

δ = min{di }. If Δ = δ, then G is called regular. The adjacency tensor [6] of a k-uniform i

hypergraph G, denoted by AG , is an order k dimension |V (G)| nonnegative tensor with entries ⎧ 1 ⎨ , if { i1 , i2 , . . . , ik } ∈ E (G) , (k − 1)! ai1 i2 ···ik = ⎩ 0, otherwise. Eigenvalues of AG are called eigenvalues of G, the spectral radius of AG is called the spectral radius of G, denoted by ρ(G). For k-uniform hypergraph G with n vertices, it is connected if and only if AG is nonnegative weakly irreducible [8,21,27]. By the Perron–Frobenius theorem [27], ρ (G) T is an eigenvalue of G and there exists a unique positive eigenvector x = (x1 , . . . , xn ) n  k corresponding to ρ (G) with xi = 1, which is called the principal eigenvector of G. i=1

The maximum and minimum entries of x are denoted by xmax and xmin , respectively. The parameter γ = xxmax is called the principal ratio of G (see [17]). For e = {i1 , i2 , . . . , ik } ∈ min k k xij and xe\{i1 } = xij . E(G), let xe = j=1

j=2

The principal eigenvector of a connected graph is one of centrality metrics [11]. Let G be a connected irregular graph. Investigations have been conducted to the relationship between xmax and the structure of a connected graph [20], bounds on γ, xmax , xmin

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and ρ(G) of G were presented in [4,29]. A connected graph G is regular if and only if ρ(G) = Δ and γ = 1 (see [29]). For a connected irregular graph G, some bounds on Δ − ρ(G) were given in [3,5,16,23,24,29]. Recently, the principal eigenvector of a uniform hypergraph has attracted attention. Let G be a connected uniform hypergraph. Nikiforov [19] gave some analytic methods for studying uniform hypergraphs, and presented some bounds on entries for the principal eigenvector of G (see Section 7 of [19]). Liu et al. [17] investigated some bounds on entries for the principal eigenvector of G, and gave an estimate of the gap of spectral radii between G and its proper sub-hypergraph G . In addition, many researchers were interested in the spectral radii of hypergraphs, and gave some bounds to estimate the spectral radius (see [2,6,9,10,12,15,17,19,21,25,28,30]). This paper is organized as follows. In Section 2, some auxiliary lemmas are presented. In Section 3, some inequalities among the principal eigenvector, spectral radius and vertex degrees of a connected uniform hypergraph are established. Necessary and sufficient conditions of equalities holding are presented, which are related to the regularity of a hypergraph. In Section 4, for a connected irregular uniform hypergraph G, we obtain some lower bounds on Δ − ρ(G) in terms of some parameters, such as principal ratio, maximum degree, diameter, and the number of vertices and edges. 2. Preliminaries In this section, some helpful lemmas are presented. Lemma 2.1. [13] Let y1 , . . . , yn be nonnegative numbers (n ≥ 2). Then 1 y1 + · · · + yn 1 − (y1 · · · yn ) n  n n(n − 1)



√ √ ( yi − yj )2 ,

1i
equality holds if and only if y1 = · · · = yn . Lemma 2.2. [23] Let a, b, y1 , y2 be positive numbers. Then a(y1 − y2 )2 + by22  equality holds if and only if y2 =

ab 2 y , a+b 1

ay1 a+b .

Lemma 2.3. [26] Let A be a nonnegative tensor, and λ be an eigenvalue of A. If there exists a positive eigenvector corresponding to λ, then λ must be ρ(A). π Lemma 2.4. [7] Let Pn be a path with n vertices. Then ρ(Pn ) = 2 cos n+1 .

Let G = (V, E) be a graph. For any k ≥ 3 and 1 ≤ s ≤ k2 , the generalized power [9] of G, denoted by Gk,s , is defined as the k-uniform hypergraph with the vertex set

276

H. Li et al. / Linear Algebra and its Applications 544 (2018) 273–285

{v : v ∈ V } ∪ {e : e ∈ E}, and edge set {u ∪ v ∪ e : e = {u, v} ∈ E}, where v is an s-set containing v and e is a (k − 2s)-set corresponding to e. k

Lemma 2.5. [9] Let G = (V, E) be a graph. Then ρ(G) = ρ(Gk, 2 ). 3. The principal eigenvectors of hypergraphs For a connected graph G, it is regular if and only if the principal ratio γ = 1, so one can regard γ as a measure of the irregularity for G (see [4]). For a connected k-uniform hypergraph G with n vertices, if G is regular, then ρ(G) = Δ (see [6]), and

1 T 1 there exists a vector x = n− k , . . . , n− k such that AG xk−1 = ρ(G)x[k−1] . Hence, x is the principal eigenvector of G, so γ = 1. Conversely, if γ = 1, then the princi 1 T 1 pal eigenvector x = n− k , . . . , n− k . For all i ∈ [n], by (AG xk−1 )i = ρ(G)xk−1 , i.e. i  xe\{i} = ρ (G) xk−1 , we obtain ρ(G) = di , G is regular. Thus, G is regular if and i e∈Ei (G)

only if γ = 1.

For a connected graph G, Zhang [29] showed that γ ≥ Δ δ , we extend it to hypergraph as (i) in Theorem 3.1, and analyze the condition of equality holding. Cioabˇ a et al. [4] proved that xmax ≥ 1Δ−δ , equality holds if and only if G is regular, we extend it to n−

Δ

hypergraph as (ii) in Theorem 3.1. Theorem 3.1. For a connected k-uniform hypergraph G with n vertices (k > 2), let γ and x be the principal ratio and principal eigenvector of G, respectively. Then 1   2(k−1) (i) γ ≥ Δ , equality holds if and only if xp = xmax , xs = xmin , xq = xmin , δ xt = xmax , for all p ∈ {i : di = Δ, i ∈ V (G)}, q ∈ {i : di = δ, i ∈ V (G)}, s ∈ Ep (G)\{p}, and t ∈ Eq (G)\{q}.

  k − k1 δ 2(k−1) (ii) xmax ≥ Δ +n−1 , equality holds if and only if G is regular. 1

 − k k  Δ  2(k−1) (iii) xmin ≤ +n−1 , equality holds if and only if G is regular. δ Proof. (i) From AG xk−1 = ρ (G) x[k−1] , for all p ∈ {i : di = Δ, i ∈ V (G)}, q ∈ {i : di = δ, i ∈ V (G)}, we have ρ (G) xk−1 = p



xe\{p} ≥ Δxk−1 min ,

(3.1)

xe\{q} ≤ δxk−1 max .

(3.2)

e∈Ep (G)

and ρ (G) xk−1 = q

 e∈Eq (G)

It follows that

H. Li et al. / Linear Algebra and its Applications 544 (2018) 273–285

xk−1 Δxk−1 xk−1 p max min ≥ ≥ , k−1 k−1 xk−1 x δx q max min 1

 2(k−1) xmax Δ γ= ≥ . xmin δ

277

(3.3)

1   2(k−1) By (3.1)–(3.3), we know that γ = Δ if and only if xp = xmax , xs = xmin , δ xq = xmin , xt = xmax , for all p ∈ {i : di = Δ, i ∈ V (G)}, q ∈ {i : di = δ, i ∈ V (G)}, s ∈ Ep (G)\{p}, and t ∈ Eq (G)\{q}. (ii) By (i), it is easy to see

1=

n 

xki ≤ xkmin + (n − 1) xkmax

i=1

(3.4)

  = xkmax γ −k + n − 1    k δ 2(k−1) k ≤ xmax +n−1 , Δ

(3.5) (3.6)

then  xmax ≥

δ Δ

k  2(k−1)

− k1 +n−1

.



  k δ 2(k−1) If the equality in (ii) holds, then xkmax Δ + n − 1 = 1, we obtain the equalities in (3.4) and (3.6) hold, it implies that n − 1 components of x equal to xmax and one 1 1   2(k−1)   2(k−1) component of x equal to xmin , and γ = Δ . If γ = Δ , then xp = xmax , xs = δ δ xmin , for all p ∈ {i : di = Δ, i ∈ V (G)}, s ∈ Ep (G)\{p}. Since one component of x equal to xmin and k > 2, we can get xmax = xmin , G is regular. Conversely, if G is regular, then

1 − k1 T

  k 1 δ 2(k−1) the principal eigenvector x = n− k , . . . , n− k , we can get + n − 1 = Δ n− k , the equality in (ii) holds. (iii) From (i), we have 1

1=

n  i=1

then

xki ≥ xkmax + (n − 1) xkmin   = xkmin γ k + n − 1   k  2(k−1) Δ +n−1 , ≥ xkmin δ

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278

 xmin ≤

Δ δ

k  2(k−1)

− k1 +n−1

.

Similarly to the proof in (ii), we obtain that (iii) holds. 2 1   2(k−1) Remark 1. The hypergraph satisfying γ = Δ may not be regular. Now we present δ a connected irregular uniform hypergraph such that the equality in (i) holds. Let G be a connected 3-uniform hypergraph with V (G) = {1, 2, . . . , 30} and E(G) = {ei : 1 ≤ i ≤ 64}, where

e1 e5 e9 e13 e17 e21 e25 e29 e33 e37 e41 e45 e49 e53 e57 e61

= {1, 2, 5}, = {1, 2, 9}, = {3, 4, 6}, = {3, 4, 10}, = {5, 10, 11}, = {7, 8, 11}, = {6, 8, 9}, = {1, 2, 12}, = {3, 4, 13}, = {15, 16, 17}, = {15, 18, 20}, = {15, 19, 24}, = {15, 20, 28}, = {15, 22, 23}, = {15, 24, 27}, = {15, 26, 27},

e2 e6 e10 e14 e18 e22 e26 e30 e34 e38 e42 e46 e50 e54 e58 e62

= {1, 2, 6}, = {1, 2, 10}, = {3, 4, 7}, = {3, 4, 11}, = {5, 6, 8}, = {7, 9, 11}, = {6, 9, 10}, = {3, 4, 12}, = {1, 2, 14}, = {15, 16, 29}, = {15, 18, 21}, = {15, 19, 25}, = {15, 21, 29}, = {15, 22, 24}, = {15, 24, 28}, = {15, 26, 28},

e3 e7 e11 e15 e19 e23 e27 e31 e35 e39 e43 e47 e51 e55 e59 e63

= {1, 2, 7}, = {1, 2, 11}, = {3, 4, 8}, = {5, 6, 7}, = {5, 7, 10}, = {7, 8, 10}, = {6, 10, 11}, = {13, 15, 16}, = {14, 15, 16}, = {15, 17, 30}, = {15, 18, 22}, = {15, 20, 26}, = {15, 21, 30}, = {15, 23, 25}, = {15, 25, 29}, = {15, 27, 29},

e4 e8 e12 e16 e20 e24 e28 e32 e36 e40 e44 e48 e52 e56 e60 e64

= {1, 2, 8}, = {3, 4, 5}, = {3, 4, 9}, = {5, 8, 9}, = {5, 9, 11}, = {6, 7, 8}, = {9, 10, 11}, = {13, 15, 17}, = {14, 15, 17}, = {15, 18, 19}, = {15, 19, 23}, = {15, 20, 27}, = {15, 21, 22}, = {15, 23, 26}, = {15, 25, 30}, = {15, 28, 30}.

By direct calculation, we have Δ = 32 and δ = 2. For G, there exists a vector x = 1 √ (2, . . . , 2, 1, 1, 1, 2, 1, . . . , 1)T such that AG x2 = 8x[2] . Since AG ≥ 0 and x > 0, 3 114       11

15

by Lemma 2.3, we can obtain 8 is the spectral radius of AG , and x is the principal  1/4 eigenvector, γ = xxmax =2= Δ . The equality in (i) holds. δ min Remark 2. When k = 2, G may not be regular in the case of equality in (iii) holds. We present a connected irregular uniform hypergraph such that the equality in (iii) holds. Let G be a graph with V (G) = {1, 2, 3, 4} and E(G) = {e1 , e2 , e3 }, where e1 = {1, 2}, e2 = {1, 3}, e3 = {1, 4}, By direct calculation, we have Δ = 3, δ = 1, and the principal √   − 12 eigenvector x = √16 ( 3, 1, 1, 1)T , so Δ = √16 = xmin , equality in (iii) δ +n−1 holds.

H. Li et al. / Linear Algebra and its Applications 544 (2018) 273–285

For a connected graph G, Cioabˇ a et al. [4] showed that xmax ≥

ρ(G) , n  d2i

279

and we extend

i=1

it to hypergraph as (ii) in Theorem 3.2. Theorem 3.2. For a connected k-uniform hypergraph G with n vertices and m edges. Let ρ(G) and x be the spectral radius and the principal eigenvector of G, respectively. Then

 k1 (i) xmax ≥ ρ(G) , equality holds if and only if G is regular. km 1

(ii) xmax ≥



ρ(G) k−1 1 k k n  dik−1

, equality holds if and only if G is regular.

i=1

Proof. (i) From the equation AG xk−1 = ρ(G)x[k−1] , we have    ρ (G) = xT AG xk−1 = k xe ≤ kmxkmax . e∈E(G)

Thus,

xmax ≥

ρ (G) km

 k1 .

 k1 Clearly, xmax = ρ(G) if and only if xi = xmax for all i ∈ V (G), it follows that G is km regular. (ii) From AG xk−1 = ρ (G) x[k−1] , for all i ∈ V (G), we have 

ρ (G) xk−1 = i

xe\{i} ≤ di xk−1 max ,

e∈Ei (G) k

ρ(G) k−1

n 

xki ≤ xkmax

i=1

Since

n  i=1

n 

k

dik−1 .

i=1 k

xi k = 1, we obtain xkmax ≥

ρ(G) k−1 k n  dik−1

, i.e.

i=1

1

xmax

ρ(G) k−1 ≥  k1 . n k  k−1 di i=1

1

Clearly, xmax = is regular. 2



ρ(G) k−1 1 k k n  dik−1

i=1

if and only if xi = xmax for all i ∈ V (G), it follows that G

H. Li et al. / Linear Algebra and its Applications 544 (2018) 273–285

280

⎛

n 

k

dik−1

Remark 3. When ρ(G) ≤ ⎝ i=1km

ρ(G) km

 k1

⎞k−1 ⎠

, by Theorem 3.2, it yields that xmax ≥ ⎛

≥



1 ρ(G) k−1 n  i=1

k dik−1

k

dik−1

. Conversely, when ρ(G) > ⎝ i=1km

1

k



1

we can obtain xmax ≥

n 



ρ(G) k−1 1 k k n  dik−1

>

ρ(G) km

 k1

⎞k−1 ⎠

, by Theorem 3.2,

.

i=1

4. The spectral radii of hypergraphs Let G be a uniform hypergraph. A path P of G is an alternating sequence of vertices and edges v0 e1 v1 e2 · · · vl−1 el vl , where v0 , . . . , vl are distinct vertices of G, e1 , . . . , el are distinct edges of G and vi−1 , vi ∈ ei , for i = 1, . . . , l. The number of edges in P is called the length of P . The distance d(u, v) between two distinct vertices u and v of G is the length of the shortest path connecting them. The diameter D of G is the maximum distance among all vertices of G. For a connected irregular graph G with n vertices and m edges, Cioabˇ a et al. [5] proved nΔ−2m that Δ − ρ(G) > n(D(nΔ−2m)+1) . In the following, some lower bounds on Δ − ρ(G) for a connected irregular uniform hypergraph G are presented. Theorem 4.1. Let G be a connected irregular k-uniform hypergraph with n vertices and m edges (k ≥ 3). Then Δ − ρ(G) >

Δ(nΔ − km) . 2m(k − 1)D(nΔ − km) + nΔ

T

Proof. Let x = (x1 , . . . , xn ) be the principal eigenvector of G. Suppose that u, v ∈ V (G) such that u = v, xu = xmax and xv = xmin . Then Δ − ρ(G) =

n 

(Δ −

di )xki

+

i=1

=

n 

n 

di xki − k

i=1

xe

e∈E(G)



(Δ − di ) xki +

i=1



 k  xi1 + · · · + xkik − kxe .

{i1 ,...,ik }=e∈E(G)

Since G is irregular, we have n 

(Δ − di ) xki >

i=1

By Lemma 2.1, it yields that

n  i=1

(Δ − di ) xkv = (nΔ − km)xkv .

H. Li et al. / Linear Algebra and its Applications 544 (2018) 273–285



 k  xi1 + · · · + xkik − kxe >

1 k−1

{i1 ,...,ik }=e∈E(G)

281

k 2 k xr2 − xs2 .

 {r,s}⊂e∈E(G)

Hence we obtain Δ − ρ(G) =

n 



(Δ − di ) xki +



xki1 + · · · + xkik − kxe



{i1 ,...,ik }=e∈E(G)

i=1

> (nΔ − km)xkv +

1 k−1





k

k

xr2 − xs2

2 .

(4.1)

{r,s}⊂e∈E(G)

Let P = v0 e1 v1 e2 · · · vl−1 el vl be the shortest path from vertex u to vertex v, where u = v0 , v = vl . Then



k

k

xr2 − xs2

2

{r,s}⊂e∈E(G)

≥

l 

k 2 vi−1

⎝ x

−x

k 2 vi

2

≥

⎛



k 2 vi−1

⎝ x

−x

k 2 vi

2

l



x

k 2 vi−1

−x

k 2 vi

2

i=1

=

k

2

x

k 2 vi−1

−x

k 2 uj

2

⎞

k 2  k ⎠ + xu2j − xv2i

uj ∈ei \{vi−1 ,vi }

+

i=1

=





+

i=1 l 

k

xr2 − xs2

{r,s}⊂e∈E(P )

⎛







≥

1 2





k 2 vi−1

x

−x

k 2 uj

+x

k 2 uj

−x

k 2 vi

2

⎞ ⎠

uj ∈ei \{vi−1 ,vi }

2 k k − 2 k2 xvi−1 − xv2i + 2



l 2 k k  k2 xvi−1 − xv2i . 2 i=1

By Cauchy–Schwarz inequality, we obtain 



k 2

k 2

xr − x s

{r,s}⊂e∈E(G)

2

k ≥ 2l



l

  k k xv2i−1 − xv2i

2

i=1

2 2 k k k k2 k k2 xu − xv2 xu − xv2 . = ≥ 2l 2D

(4.2)

From (4.1) and (4.2), it yields that Δ − ρ(G) > (nΔ − km) xkv + By Lemma 2.2, we can get

2

k k k xu2 − xv2 . 2(k − 1)D

(4.3)

H. Li et al. / Linear Algebra and its Applications 544 (2018) 273–285

282

Δ − ρ(G) >

k(nΔ − km) xk . 2(k − 1)D(nΔ − km) + k u

By (i) of Theorem 3.2, we have Δ − ρ(G) >

(nΔ − km)ρ(G) k(nΔ − km)ρ(G) = (2(k − 1)D(nΔ − km) + k) km m (2(k − 1)D(nΔ − km) + k)

=

(nΔ − km)Δ (nΔ − km)(Δ − ρ(G)) − . m (2(k − 1)D(nΔ − km) + k) m (2(k − 1)D(nΔ − km) + k)

Thus, Δ − ρ(G) >

Δ(nΔ − km) . 2 2m(k − 1)D(nΔ − km) + nΔ

Remark 4. From Theorem 4.1, we have Δ − ρ(G) >

Δ(nΔ − km) = 2m(k − 1)D(nΔ − km) + nΔ

1 2m(k−1)D Δ

Since G is irregular, we can get that nΔ − km = nΔ − n 

n 

+

n nΔ−km

.

di ≥ 1, (k − 1)m < km =

i=1

di < nΔ, then

i=1

Δ − ρ(G) >

1 2m(k−1)D Δ

+

n nΔ−km

>

2 1 ≥ . n(2D + 1) 5nD

(1)

The following hypergraph suggests that the lower bound in Theorem 4.1 is asymptotically k, k best possible. Let Pn be a path with n vertices (n > 2). Then Pn 2 is an irregular k, k



2 k-uniform hypergraph with n = kn is D = n − 1, and 2 vertices, the diameter of Pn Δ = 2, δ = 1. By Lemma 2.4 and Lemma 2.5, we obtain

k, k

ρ(Pn 2 ) = ρ(Pn ) = 2 cos

π . n + 1

Hence we have k, k

Δ − ρ(Pn 2 ) = 2 − 2 cos =  2n k

 2 π π π = 2 1 − cos < n + 1 n + 1 n + 1

kπ 2 π2  < . 2nD + 1 (D + 2) k, k

When k is fixed, the upper bound of Δ − ρ(Pn 2 ) has the same order of magnitude as the lower bound in (1).

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283

Proposition 4.2. Let G be a connected irregular k-uniform hypergraph with n vertices and m edges. Then Δ − ρ(G) > max{Φ1 , Φ2 }, where

2  k Δ 2(k − 1)D(nΔ − km)γ −k + k 1 − γ − 2 Φ1 =

2 , k 2(k − 1)D (km + (nΔ − km)γ −k ) + k 1 − γ − 2

Φ2 =

2 k 2(k − 1)D(nΔ − km)γ −k + k 1 − γ − 2 2 (γ −k + n − 1) (k − 1)D

.

T

Proof. Let x = (x1 , . . . , xn ) be the principal eigenvector of G. Suppose that u, v ∈ V (G) such that u = v, xu = xmax and xv = xmin . From (4.3), we can get 2

k k k xu2 − xv2 2(k − 1)D

2  k k −k −k 2 . 1−γ = xu (nΔ − km)γ + 2(k − 1)D

Δ − ρ(G) > (nΔ − km) xkv +

(4.4)

It follows from (i) of Theorem 3.2 that

2  k ρ(G) k (nΔ − km)γ −k + 1 − γ− 2 km 2(k − 1)D

2 

Δ k −k −k = (nΔ − km)γ + 1−γ 2 km 2(k − 1)D

2  Δ − ρ(G) k −k −k 2 − , (nΔ − km)γ + 1−γ km 2(k − 1)D

Δ − ρ(G) >

so we have

2  −k 2 Δ 2(k − 1)D(nΔ − km)γ + k 1 − γ Δ − ρ(G) >

2 = Φ1 . k 2(k − 1)D (km + (nΔ − km)γ −k ) + k 1 − γ − 2

−k

By (4.4) and (3.5), we get

Δ − ρ(G) >

(nΔ − km)γ −k +

k 2(k−1)D

γ −k + n − 1



1 − γ− 2

k

2

(4.5)

284

H. Li et al. / Linear Algebra and its Applications 544 (2018) 273–285

=

2 k 2(k − 1)D(nΔ − km)γ −k + k 1 − γ − 2 2 (γ −k + n − 1) (k − 1)D

= Φ2 .

(4.6)

From (4.5) and (4.6), we obtain Δ − ρ(G) > max{Φ1 , Φ2 }.

2

Remark 5. We present two examples to compare Φ1 with Φ2 in Theorem 4.2. Let G1 be a 3-uniform hypergraph with vertex set V (G1 ) = {1, 2, 3, 4, 5} and edge set E(G1 ) = {e1 , e2 }, where e1 = {1, 2, 3}, e2 = {1, 4, 5}. By Ng–Qi–Zhou algorithm [18], we obtain γ ≈ 1.2599. From Theorem 4.2, we have Φ1 ≈ 0.5060, Φ2 ≈ 0.4516, then Φ2 < Φ1 . Let G2 be a 3-uniform hypergraph with vertex set V (G2 ) = {1, 2, 3, 4, 5} and edge set E(G2 ) = {e1 , e2 , e3 , e4 }, where e1 = {1, 2, 3}, e2 = {1, 3, 4}, e3 = {1, 4, 5}, e4 = {2, 4, 5}. By Ng–Qi–Zhou algorithm [18], we obtain γ ≈ 1.1915. From Theorem 4.2, we have Φ1 ≈ 0.3901, Φ2 ≈ 0.3907, then Φ2 > Φ1 . Acknowledgements The authors are thankful to Prof. R.A. Brualdi and reviewers for their valuable comments and suggestions towards improving the original version of this paper. This work is supported by the National Natural Science Foundation of China (No. 11371109, No. 11671108 and No. 11601102), the Fundamental Research Funds for the Central Universities (No. GK2110260149). References [1] C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. [2] C. Bu, X. Jin, H. Li, C. Deng, Brauer-type eigenvalue inclusion sets and the spectral radius of tensors, Linear Algebra Appl. 512 (2017) 234–248. [3] S.M. Cioabˇ a, The spectral radius and the maximum degree of irregular graphs, Electron. J. Combin. 14 (2007) R38. [4] S.M. Cioabˇ a, D.A. Gregory, Principal eigenvectors of irregular graphs, Electron. J. Linear Algebra 16 (2007) 366–379. [5] S.M. Cioabˇ a, D.A. Gregory, V. Nikiforov, Extreme eigenvalues of nonregular graphs, J. Combin. Theory Ser. B 97 (2007) 483–486. [6] J. Cooper, A. Dutle, Spectra of uniform hypergraphs, Linear Algebra Appl. 436 (2012) 3268–3292. [7] D. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, Cambridge, 2010. [8] S. Friedland, S. Gaubert, L. Han, Perron–Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl. 438 (2013) 738–749. [9] M. Khan, Y. Fan, On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs, Linear Algebra Appl. 480 (2015) 93–106. [10] M. Khan, Y. Fan, Y. Tan, The H-spectra of a class of generalized power hypergraphs, Discrete Math. 339 (2016) 1682–1689. [11] C. Li, H. Wang, P.V. Mieghem, New lower bounds for the fundamental weight of the principal eigenvector in complex networks, in: 2014 Tenth International Conference on Signal-Image Technology and Internet-Based Systems, IEEE, 2014, pp. 317–322.

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