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Sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor and its application

South China Normal University, Lihua YOU Joint work with Xiaohua HUANG, Xiying YUAN, Yujie SHU, Pingzhi YUAN July 7, 2018

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Outline of the Talk

1

Part I : Matrix and its spectrum

2

Part II : Tensor and its spectrum

3

Part III : Applications to a k-uniform hypergraph

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Part I : Matrix and its spectrum

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Definitions and notations

M: a real matrix of order n. λ1, λ2, . . . , λn: the eigenvalues of M with |λ1| ≥ |λ2| ≥ . . . ≥ |λn|. ρ(M): the spectral radius of M, ρ(M) = |λ1|. If M is a nonnegative matrix, it follows from the Perron-Frobenius theorem that the spectral radius ρ(M) is a eigenvalue of M. If M is a nonnegative irreducible matrix, it follows from the Perron-Frobenius theorem that ρ(M) = λ1 is simple.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Definitions and notations

Let G be a graph. A(G) = (aij): the adjacency matrix of G, where aij = 1 if vi and vj are adjacent and 0 otherwise. The spectral radius of A(G) : ρ(G). diag(G) = diag(d1, d2, . . . , dn): the diagonal matrix of vertex degrees of a graph G. The signless Laplacian matrix of G: Q(G) = diag(G) + A(G). The spectral radius of Q(G) : q(G).

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Definitions and notations

D(G) = (dij): the distance adjacency matrix of G. The spectral radius of D(G) : ρD(G). Tr(G) = diag(D1, D2, . . . , Dn): the diagonal matrix of vertex transmission of G. The distance signless Laplacian matrix of G: Q(G) = Tr(G) + D(G). The spectral radius of Q(G) : qD(G).

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Definitions and notations

Let − → G be a digraph. A(− → G) = (aij): the adjacency matrix of − → G, where aij is equal to the number of arc (vi, vj). The spectral radius of A(− → G) : ρ(− → G). diag(− → G) = diag(d+

1 , d+ 2 , . . . , d+ n): the diagonal matrix of

vertex out-degrees of − → G. The signless Laplacian matrix of − → G: Q(− → G) = diag(− → G) + A(− → G). The spectral radius of Q(− → G) : q(− → G).

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Notation of a graph

Let G = (V, E) be a simple (connected) graph with vertex set V = V(G) = {v1, v2, . . . , vn} and edge set E = E(G). di: the degree of vertex vi. i ∼ j: vi is adjacent to vj. mi =

• i∼j

dj di : the average degree of the neighbors of vi in G.

duv: the distance between u and v, is the length of the shortest path connecting them in G. Di =

n

• j=1

dij: the distance degree of vertex vi in G, or the transmission of vertex vi in G. Ti =

n

• j=1

dijDj: the second distance degree of vertex vi in G.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Some known results

Theorem 1 Let G = (V, E) be a simple connected graph on n vertices. Then

ρ(G) ≤ max

1≤i,j≤n

√mimj, i ∼ j

• .

Moreover, the equality holds if and only if one of the following two conditions holds: (1) m1 = m2 = . . . = mn; (2) G is a bipartite graph and the vertices of same partition have the same average degree.

K.C. Das, P. Kumar, Some new bounds on the spectral radius

• f graphs, Discrete Math. 281 (2004) 149–161.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Some known results

Theorem 2 Let G = (V, E) be a connected graph on n vertices, for any 1 ≤ i, j ≤ n, g(i, j) =

di+dj+√ (di−dj)2+4mimj 2

. Then we have min{g(i, j), i ∼ j} ≤ q(G) ≤ max{g(i, j), i ∼ j}, and one of the equalities holds if and only if one of the following conditions holds: (1) G is a regular graph; (2) G is a bipartite semi-regular graph; A.D. Maden, K.C. Das, A.S. Cevik, Sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph, Appl Math Comput. 219 (2013) 5025–5032.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Some known results

Theorem 3 Let G = (V, E) be a connected graph on n vertices. Then ρD(G) ≤ max

1≤i,j≤n

• TiTj

DiDj

• ,

and the equality holds if and only if

T1 D1 = T2 D2 = . . . = Tn Dn.

C.X. He, Y. Liu, Z.H. Zhao, Some new sharp bounds on the distance spectral radius of graph, MATCH Commum. Math.

• Comput. Chem. 63 (2010) 783–788.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Some known results

Theorem 4 (Hong and Y., 2014, AMC) Let G = (V, E) be a connected graph on n vertices, for all 1 ≤ i, j ≤ n, h(i, j) =

Di+Dj+

• (Di−Dj)2+

4TiTj DiDj

2

. Then qD(G) ≤ max

1≤i,j≤n{h(i, j)},

and the equality holds if and only if D1 + T1

D1 = . . . = Dn + Tn Dn .

W.X. Hong, L.H. You, Futher results on the spectral radius of matrices and graphs, Applied Math Comput. 239 (2014) 326–332.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Notation of a digraph

Let − → G = (V, E) be a digraph on n vertices. i ∼ j: (vi, vj) ∈ E. d+

i : the out-degree of the vertex vi in −

→ G. m+

i =

• i∼j

d+

j

d+

i

: the average out-degree of the out-neighbors of vi in − → G.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Some known results

Theorem 5 Let − → G = (V, E) be a strong connected digraph on n vertices. Then min

1≤i,j≤n

• m+

i m+ j , i ∼ j

• ≤ ρ(−

→ G) ≤ max

1≤i,j≤n{

• m+

i m+ j , i ∼ j

• ,

and one of the equalities holds if and only if one of the following two conditions holds: (1) m+

1 = m+ 2 = . . . = m+ n,

(2) − → G is a bipartite graph and the vertices of same partition have the same average out-degree. G.H. Xu, C.Q. Xu, Sharp bounds for the spectral radius of digraphs, Linear Algebra Appl. 430 (2009) 1607–1612.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Some known results

Theorem 6 Let − → G = (V, E) be a strong connected digraph on n vertices, G(i, j) =

d+

i +d+ j +√

(d+

i −d+ j )2+4m+ i m+ j

2

for any i, j ∈ {1, 2, . . . , n}. Then min

1≤i,j≤n{G(i, j), i ∼ j} ≤ q(−

→ G) ≤ max

1≤i,j≤n{G(i, j), i ∼ j}.

S.B. Bozkurt, D. Bozkurt, On the signless Laplacian spectral radius of digraphs, Ars Combin. 108 (2013) 193–200.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Our results

Theorem 7 (Y., Shu and Yuan, 2017, LAMA) Let A = (aij) be an n × n nonnegative irreducible matrix with aii = 0 for i = 1, 2, . . . , n, and the row sum r1, r2, . . . , rn. Let B = A + M, where M = diag(t1, t2, . . . , tn) with ti ≥ 0 for any i ∈ {1, 2, . . . , n}, si =

n

• j=1

aijrj, ρ(B) be the spectral radius of B. Let f(i, j) =

ti+tj+

• (ti−tj)2+

4sisj rirj

2

for any 1 ≤ i, j ≤ n. Then min

1≤i,j≤n{f(i, j), aij = 0} ≤ ρ(B) ≤ max 1≤i,j≤n{f(i, j), aij = 0}.

(1) L.H. You, Y.J. Shu, P.Z. Yuan, Sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and its applications, Linear Multilinear Algebra, 65(1) (2017), 113–128.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

(Cont.) Moreover, the equalities in (1) hold if and only if one of the two conditions holds: (i) ti + si

ri = tj + sj rj for any i, j ∈ {1, 2, . . . , n};

(ii) There exist nonempty proper subsets U and W of [n] such that 1) [n] = U ∪ W with U ∩ W = φ, 2) aij = 0 only when i ∈ U, j ∈ W or i ∈ W, j ∈ U, 3) there exists ℓ > 0 such that ρ(B) = ti + ℓsi

ri = tj + sj ℓrj for all

i ∈ U and j ∈ W. In fact, ℓ > 1 when the left equality holds and ℓ < 1 when the right equality holds.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Part II : Tensor and its spectrum

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Tensor (or hypermatrix, or multiple array) Definition An order m dimension n tensor (or hypermatrix, or multiple array) T = (ti1i2···im) (1 ≤ ij ≤ n, j = 1, · · · , m) over the complex field C is a multidimensional array with all entries ti1i2···im ∈ C.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

A general product of tensors Let A (and B) be order m ≥ 2 (and k ≥ 1) dimension n tensor,

• respectively. The product AB is the following tensor C of

dimension n and order (m − 1)(k − 1) + 1 with entries: Ciα1···αm−1 =

• i2,··· ,im∈[n2]

Aii2···imBi2α1 · · · Bimαm−1, where i ∈ [n], α1, · · · , αm−1 ∈ [n]k−1.

• J. Shao, A general product of tensors with applications, Linear

Algebra Appl., 439(2013)2350-2366.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Eigenequations, eigenvalues and eigenvectors of a tensor Let T be an order m dimension n tensor, and x = (x1, · · ·, xn)T ∈ Cn be a column vector of dimension n. By the rules of general product of tensors defined by Shao, T x is a vector in Cn whose ith component is as the following (T x)i =

n

• i2,··· ,im=1

tii2···imxi2 · · · xim.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Eigenequations, eigenvalue and eigenvector Denote by x[r] = (xr

1, · · · , xr n)T.

A number λ ∈ C is called an eigenvalue of the m-th order tensor T if there exists a nonzero vector x ∈ Cn satisfying the following eigenequations T x = λx[m−1], and in this case, x is called an eigenvector of T corresponding to eigenvalue λ. K.C. Chang, K. Pearson, T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci. 6(2008)507-520.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

The spectral radius for a tensor

Let T be an order m dimension n tensor. ♯ eigenvalues of T = n(m − 1)n−1. The spectral radius of T : ρ(T ) = max{|µ| : µ is an eigenvalue of T }. K.C. Chang, K. Pearson, T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Comm. Math. Sci., 2008, 6 (2): 507-520. L.Q. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput., 2005, 40: 1302-1324.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Sharp upper and lower bounds

Theorem 8 (Y., Huang and Yuan, 2018+) Let A = (ai1i2···im) be a nonnegative weakly irreducible tensor with

• rder m dimension n and ai···i = 0 for i ∈ [n]. Let ti ≥ 0, Ri > 0,

and Si =

n

• i2,...,im=1

aii2...imRi2 . . . Rim for any i ∈ [n]. Let B = A + M, where M is a diagonal tensor with its diagonal element mii···i being ti. For any 1 ≤ i, j ≤ n, write F(i, j) = ti + tj +

• (ti − tj)2 +

4SiSj (RiRj)m−1

2 . Then min

1≤i,j≤n{F(i, j), j ∈ N(i)} ≤ ρ(B) ≤ max 1≤i,j≤n{F(i, j), j ∈ N(i)}.

(2)

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

(Cont.)Moreover, one of the equalities in (2) holds if and only if

• ne of the two conditions holds:

(i) ti +

Si Rm−1

i

= tj +

Sj Rm−1

j

for any i, j ∈ [n]. (ii) There exist nonempty proper subsets U and W of [n] such that 1) [n] = U ∪ W with U ∩ W = φ; 2) ai1i2...im = 0 only when i1 ∈ U, i2, . . . , im ∈ W or i1 ∈ W, i2, . . . , im ∈ U; 3) there exists ℓ > 0 such that ρ(B) = ti + ℓm−1Si

Rm−1

i

= tj +

Sj ℓm−1Rm−1

j

for all i ∈ U and j ∈ W. In fact, ℓ > 1 when the left equality holds and ℓ < 1 when the right equality holds.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Applications to tensors and matrices

• The i-th row sum of A is defined as ri(A) =

n

• i2,··· ,im=1

aii2...im.

• Denote NA(i) (or simply N(i)) by

NA(i) = {i2, . . . , im|aii2...im = 0}.

• Write si =

n

• i2,...,im=1

aii2...imri2 . . . rim. Application 1: Let Ri = ri and Si = si. Application 2: Let m = 2, Ri = ri and Si = si. Then Theorem 7 is the corollary of Theorem 8.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Main strategy of the proof of Theorem 8

• Two diagonal similar tensors have the same spectrum. Consider

R−(m−1)BR instead of B, where R = diag(R1, . . . , Rn). Lemma 1 Suppose that the two tensors A and B are diagonal similar, namely B = D−(m−1)AD for some invertible diagonal matrix D. Then x is an eigenvector of B corresponding to the eigenvalue λ if and only if y = Dx is an eigenvalues of A corresponding to the same eigenvalue λ.

• J. Shao, A general product of tensors with applications, Linear

Algebra Appl., 439(2013)2350-2366.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Main strategy of the proof of Theorem 8

• Analyze the components of the Perron vector of a nonnegative

weakly irreducible tensor. Lemma 2 Let A be a nonnegative tensor of order m dimension n. If some eigenvalue of A has a positive eigenvector corresponding to it, then this eigenvalue must be ρ(A). Y.N. Yang, Q.Z. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal.Appl. 31(5) (2010) 2517–2530.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Main strategy of the proof of Theorem 8

Lemma 3 Let A be a nonnegative tensor of order m dimension n. Then ρ(A) is an H-eigenvalue of ρ(A) with a nonnegagtive eigenvector. Furthermore, if A is weakly irreducible, then ρ(A) has a positive eigenvector.

• S. Friedland, A. Gaubert, L. Han, Perron-Frobenius theorems

for nonnegative multilinear forms and extensions, Linear Algebra Appl. 438 (2013), 738–749. Y.N. Yang, Q.Z. Yang, On some properties of nonnegative weakly irreducible tensors, arXiv: 1111.0713v3, 2011.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Main strategy of the proof of Theorem 8

• To prove the equalities cases, the (nonnegative matrix)

representation G(A) of a tensor A, and the associated directed graph D(G(A)) of a nonnegative matrix and its strongly connectedness will be used.

• G(A) is the representation associated with the nonnegative

tensor A, if the (i, j)-th entry of G(A) is defined to be the summation of Aii2···im with indices {i2, · · · im}, where j ∈ {i2, · · · , im}.

• A nonnegative matrix A is irreducible if and only if its associated

directed graph D(A) is strongly connected.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Main strategy of the proof of Theorem 8

Proposition 1 Let A be a nonnegative tensor of order m and dimension n, G(A) be the representation associated matrix to A, and D(G(A)) be the associated directed graph of G(A). Then the following three conditions are equivalent: (1) A is weakly irreducible. (2) G(A) is irreducible. (3) D(G(A)) is strongly connected.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Part III : Applications to a k-uniform hypergraph

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Hypergraph and k-uniform hypergraph Definition A hypergraph is a pair H = (V(H), E(H)), where V(H) is the set of vertices of H and E(H) is a family of non-empty subsets of V(H). Definition A k- uniform hypergraph (or called k-graph ) H consists of a set of vertices V(H) and a set E(H) of k-subsets of V(H).

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Tensors of uniform hypergraphs Definition The adjacency tensor of a k-uniform hypergraph H with n vertices is defined as the k-th order n-dimensional tensor A(H) with (A(H))i1i2···ik =

• 1

(k−1)!

{i1, i2, · · · , ik} ∈ E(H)

• therwise.
• J. Cooper and A. Dutle, Spectra of uniform hypergraphs,

Linear Algebra and Its Applications, 436 (2012) 3268-3292.

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Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

An example H (A(H))123 = (A(H))132 = (A(H))213 = (A(H))231 = (A(H))312 = (A(H))321 = 1 2, · · · · · · (A(H))124 = (A(H))142 = (A(H))214 = (A(H))241 = (A(H))412 = (A(H))421 = 0, · · · · · ·

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Tensors of uniform hypergraphs Definition Let D(H) be a k-th order n-dimensional diagonal tensor with its diagonal entry Dii···i being di, the degree of vertex i, for all i ∈ V(H) = [n] and the other entries being 0. Then Q(H) = D(H) + A(H) is the signless Laplacian tensor of the hypergraph H.

• L. Qi, H+-eigenvalue of Laplacian and signless Laplacian

tensors, Commun. Math. Sci. 12 (2014), 1045–1064.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Tensors of uniform hypergraphs Results It was proved that a k-uniform hypergraph H is connected if and

• nly if its adjacency tensor A(H) (and so Q(H) is weakly

irreducible. For a vertices i of k-uniform hypergraph H, denoted mi =

• {i,i2,...,ik}∈E(H)

di2...dik dk−1

i

, which is a generalization of the average

• f degrees of vertices adjacent to i of the ordinary graph.
• S. Friedland, A. Gaubert, L. Han, Perron-Frobenius theorems

for nonnegative multilinear forms and extensions, Linear Algebra Appl. 438 (2013), 738–749.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Applications of Theorem 8 to a connected k-uniform hypergraph

Take A = B = A(H) in Theorem 8. Then we have Theorem 9 Let k ≥ 3, H be a connected k-uniform hypergraph on n vertices and Ri > 0 for any i ∈ [n]. Then min

e∈E(H) min {i,j}⊆e

• R′

iR′ j ≤ ρ(A(H)) ≤ max e∈E(H) max {i,j}⊆e

• R′

iR′ j,

(3) where R′

i = R−(k−1) i

• {i,i2,...,ik}∈E(H)

Ri2 . . . Rik for any i ∈ [n]. Moreover, one of the equalities in (3) holds if and only if R′

i = R′ j

for any i, j ∈ [n].

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Applications of Theorem 8 to a connected k-uniform hypergraph

Take Ri = di. Then R′

i = mi for any i ∈ [n].

Corollary 1 Let H be a connected k-uniform hypergraph on n vertices. Then min

e∈E(H) min {i,j}⊆e

√mimj ≤ ρ(A(H)) ≤ max

e∈E(H) max {i,j}⊆e

√mimj. (4) Moreover, one of the equalities in (4) holds if and only if mi = mj for any i, j ∈ [n]. X.Y. Yuan, M. Zhang, M. Lu, Some upper bounds on the eigenvalues of uniform hypergraphs, Linear Algebra Appl. 484 (2015), 540–549.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Applications of Theorem 8 to a connected k-uniform hypergraph

Take Ri = 1. Then R′

i = di for any i ∈ [n].

Corollary 2 Let H be a connected k-uniform hypergraph on n vertices. Then min

e∈E(H) min {i,j}⊆e

• didj ≤ ρ(A(H)) ≤ max

e∈E(H) max {i,j}⊆e

• didj.

(5) Moreover, one of the equalities in (5) holds if and only if H is a regular hypergraph. X.Y. Yuan, M. Zhang, M. Lu, Some upper bounds on the eigenvalues of uniform hypergraphs, Linear Algebra Appl. 484 (2015), 540–549.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Applications of Theorem 8 to a connected k-uniform hypergraph

Take A = A(H) and B = Q(H). Then Theorem 10 Let k ≥ 3, H be a connected k-uniform hypergraph on n vertices and bi > 0 for any i ∈ [n]. Then min

e∈E(H) min {i,j}⊆e g(i, j) ≤ ρ(Q(H)) ≤ max e∈E(H) max {i,j}⊆e g(i, j),

(6) where g(i, j) =

di+dj+√ (di−dj)2+4R′

iR′ j

2

and R′

i = R−(k−1) i

• {i,i2,...,ik}∈E(H)

Ri2 . . . Rik for any i ∈ [n]. Moreover, one of the equalities in (6) holds if and only if di + R′

i = dj + R′ j for any i, j ∈ [n].

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Applications of Theorem 8 to a connected k-uniform hypergraph

Let Ri = di. Then R′

i = mi for any i ∈ [n].

Corollary 3 Let H be a connected k-uniform hypergraph on n vertices. Then min

e∈E(H) min {i,j}⊆e G(i, j) ≤ ρ(Q(H)) ≤ max e∈E(H) max {i,j}⊆e G(i, j),

(7) where G(i, j) =

di+dj+√ (di−dj)2+4mimj 2

. Moreover, one of the equalities in (7) holds if and only if di + mi = dj + mj for all i, j ∈ [n]. X.Y. Yuan, M. Zhang, M. Lu, Some upper bounds on the eigenvalues of uniform hypergraphs, Linear Algebra Appl. 484 (2015), 540–549.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Applications of Theorem 8 to a connected k-uniform hypergraph

Take Ri = 1 for each i ∈ [n]. Then R′

i = di for any i ∈ [n].

Corollary 4 Let H be a connected k-uniform hypergraph on n vertices. Then min

e∈E(H) min {i,j}⊆e(di + dj) ≤ ρ(Q(H)) ≤ max e∈E(H) max {i,j}⊆e(di + dj).

(8) Moreover, one of the equalities in (8) holds if and only if H is a regular hypergraph. X.Y. Yuan, M. Zhang, M. Lu, Some upper bounds on the eigenvalues of uniform hypergraphs, Linear Algebra Appl. 484 (2015), 540–549.

Lihua You SCNU

Part I : Matrix and its spectrum Part II : Tensor and its spectrum Part III : Applications to a k-uniform hypergraph

Thank You for Your Attention!

Lihua You SCNU