Inverse Perron values and connectivity of a uniform hypergraph

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1 Inverse Perron values and connectivity of a uniform hypergraph Changjiang Bu College of Automation College of Science Harbin Engineering University Harbin, PR China buchangjiang@hrbeu.edu.cn Jiang Zhou College of Science Harbin Engineering University Harbin, PR China zhoujiang @163.com Haifeng Li College of Automation Harbin Engineering University Harbin, PR China lihaifeng080@163.com Submitted: Oct 1, 017; Accepted: Oct 1, 018; Published: Nov, 018 c The authors. Released under the CC BY-ND license International 4.0. Abstract In this paper, we show that a uniform hypergraph G is connected if and only if one of its inverse Perron values is larger than 0. We give some bounds on the bipartition width, isoperimetric number and eccentricities of G in terms of inverse Perron values. By using the inverse Perron values, we give an estimation of the edge connectivity of a -design, and determine the explicit edge connectivity of a symmetric design. Moreover, relations between the inverse Perron values and resistance distance of a connected graph are presented. Mathematics Subject Classifications: 05C50, 05C65, 05C40, 05C1, 15A69 1 Introduction Let V G and EG denote the vertex set and edge set of a hypergraph G, respectively. G is -uniform if e = for each e EG. In particular, -uniform hypergraphs are usual graphs. For i V G, E i G denotes the set of edges containing i, and d i = E i G Corresponding author. This wor is supported by the National Natural Science Foundation of China No , No and No , the Fundamental Research Funds for the Central Universities No. GK the electronic journal of combinatorics , #P4.8 1

2 denotes the degree of i. The adjacency tensor [8] of a -uniform hypergraph G, denoted by A G, is an order dimension V G tensor with entries 1 a i i = 1!, if {i 1, i,..., i } E G, 0, otherwise. The Laplacian tensor [7] of G is L G = D G A G, where D G is the diagonal tensor of vertex degrees of G. Recently, the research on spectral hypergraph theory via tensors has attracted much attention [7-10,14,19,4]. The spectral properties of the Laplacian tensor of hypergraphs are studied in [13,5,7,9,35]. n For an order dimension n tensor T = t i i, let T x = t i i x x i. i 1,...,i =1 The algebraic connectivity of a graph plays important roles in spectral graph theory [11]. Analogue to the algebraic connectivity of a graph, Qi [7] defined the analytic connectivity of a -uniform hypergraph G as { } αg = min min j=1,...,n L G x : x R n +, n x i = 1, x j = 0 where n = V G, R n + denotes the set of nonnegative vectors of dimension n. Qi proved that G is connected if and only if αg > 0. In [0], some bounds on αg were presented in terms of degree, vertex connectivity, diameter and isoperimetric number. A feasible trust region algorithm of αg was given in [9]. For any vertex j of a -uniform hypergraph G, we define the inverse Perron value of j as { } n α j G = min L G x : x R n +, x i = 1, x j = 0. Clearly, the analytic connectivity αg = min α jg is the minimum inverse Perron j V G value. For a connected graph G, α j G is the minimum eigenvalue of L G j, where L G j is the principal submatrix of L G obtained by deleting the row and column corresponding to j. L G j is nonsingular and its inverse L G j 1 is a nonnegative matrix [16]. It is easy to see that α 1 j G is the spectral radius of L G j 1, which is called the Perron value of G. All inverse Perron values of a tree T can determine the algebraic connectivity of T [1, 15]. The resistance distance [17, 34] is a distance function on graphs. For two vertices i, j in a connected graph G, the resistance distance between i and j, denoted by r ij G, is defined to be the effective resistance between them when unit resistors are placed on every edge of G. The Kirchhoff index [17, 33] of G, denoted by KfG, is defined as the sum of resistance distances between all pairs of vertices in G, i.e., KfG = r ij G. KfG, {i,j} V G is a global robustness index. The resistance distance and Kirchhoff index in graphs have been investigated extensively in mathematical and chemical literatures [3,4,6,1,3,31,36]. the electronic journal of combinatorics , #P4.8

3 This paper is organized as follows. In Section, some auxiliary lemmas are introduced. In Section 3, we show that a uniform hypergraph G is connected if and only if one of its inverse Perron values is larger than 0, and some inequalities among the inverse Perron values, bipartition width, isoperimetric number and eccentricities of G are established. Partial results improve some bounds in [0, 7]. We also use the inverse Perron values to estimate the edge connectivity of -designs. In Section 4, some inequalities among the inverse Perron values, resistance distance and Kirchhoff index of a connected graph are presented. Preliminaries For a positive integer n, let [n] = {1,,..., n}. An order m dimension n tensor T = t i m consists of n m entries, where i j [n], j [m]. When m =, T is an n n matrix. Let R [m,n] denote the set of order m dimension n real tensors, and let R n + denote the cone of nonnegative vectors in R n. For T = t i i m R [m,n] and x = x 1,..., x n T R n, let T x m 1 R n denote the vector whose i-th component is T x m 1 i = n i,i 3,...,i m=1 t ii i m x i x i3 x im, and let x [m 1] = x m 1 1,..., x m 1 n T. In 005, Qi [6] and Lim [1] proposed the concept of eigenvalues of tensors, independently. For T = t i i m R [m,n], if there exist a number λ R and a nonzero vector x = x 1,..., x n T R n such that T x m 1 = λx [m 1], then λ is called an H-eigenvalue of T, x is called an H-eigenvector of T corresponding to λ. For a vertex j of a -uniform hypergraph G, let L G j R [,n 1] denote the principal subtensor of L G R [,n] with index set V G \ {j}. By Lemma.3 in [3], we now that α j G is the smallest H-eigenvalue of L G j for any j V G. A path P of a uniform hypergraph G is an alternating sequence of vertices and edges v 0 e 1 v 1 e v l 1 e l v l, where v 0,..., v l are distinct vertices of G, e 1,..., e l are distinct edges of G and v i 1, v i e i, for i = 1,..., l. The number of edges in P is the length of P. For all u, v V G, if there exists a path starting at u and terminating at v, then G is said to be connected [5]. Lemma 1. [7] The uniform hypergraph G is connected if and only if αg > 0. Let G be a -uniform hypergraph, S be a proper subset of V G. Denote S = V G \ S. The edge-cut set ES, S consists of edges whose vertices are in both S and S. The minimum cardinality of such an edge-cut set is called edge connectivity of G, denote by eg. Lemma. [7] Let G be a -uniform hypergraph with n vertices. Then eg n αg. the electronic journal of combinatorics , #P4.8 3

4 The {1}-inverse of a matrix M is a matrix X such that MXM = M. Let M 1 denote any {1}-inverse of M, and let M ij denote the i, j-entry of M. Lemma 3. [, 34] Let G be a connected graph. Then r ij G = L 1 G ii + L 1 G jj L 1 G ij L 1 G ji. Let tra denote the trace of the square matrix A, and let e denote an all-ones column vector. Lemma 4. [30] Let G be a connected graph of order n. Then KfG = ntrl 1 G e L 1 G e. Lemma 5. [] Let G be a connected graph with n vertices and i [n]. Let L G = L 1 x L x T d i y, where L 1 R i 1 i 1, L 3 R n i n i, x R i 1, y T R n i. L T y T L 3 L1 Suppose L G i 1 L =, where L L 1 R i 1 i 1, L 3 R n i n i. Then T L3 L 1 0 L is a symmetric {1}-inverse of L G. T L 0 L3 3 Inverse Perron values of uniform hypergraphs In the following theorem, the relationship between inverse Perron values and connectivity of a hypergraph is presented. Theorem 6. Let G be a -uniform hypergraph. Then the following statements are equivalent: 1 G is connected. α j G > 0 for all j V G. 3 α j G > 0 for some j V G. Proof. 1=. If G is connected, then by Lemma 1, we now that α j G > 0 for all j V G. = 3. Obviously. 3= 1. Suppose that G is disconnected. For any j V G, let G 1 be the component of G such that j / V G 1. Let x = x 1,..., x V G T be the vector satisfying { V G x i = 1 1, if i V G1, 0, otherwise. Clearly, we have n x i = 1. Then we have 0 α j G L G x = 0 for any j V G, a contradiction to 3. Hence G is connected if 3 holds. the electronic journal of combinatorics , #P4.8 4

5 The bipartition width of a hypergraph G is defined as [18, 8] { n } bwg = min ES, S : S V G, S =, where n denotes the maximum integer not larger than n. The computation of bwg is very difficult even for the graph case. In [], Mohar and Polja used the algebraic connectivity to obtain a lower bound on the bipartition width of a graph. In the following theorem, we use the inverse Perron values to obtain a lower bound on the bipartition width of a uniform hypergraph. Theorem 7. Let G be a -uniform hypergraph with n vertices. Then bwg n + 1n n + 1 n α j G. j=1 Proof. Suppose that S 0 V G satisfying S 0 = n and ES0, S 0 = bwg. Let x = x 1,..., x n T be the vector satisfying { S0 1, i S0, x i = 0, i S 0. Then n x i = 1. For j S 0, we get where ts 0 = 1 α j G L G x = α j G {i 1,...,i } EG {i 1,...,i } ES 0,S 0 = 1 S 0 ES 0,S 0 e ES 0,S 0 Similarly, for j S 0, we obtain Combining 1 and, we get n j=1 α j G = j S 0 α j G + j S 0 e ES 0,S 0 e S 0. x + + x i x x i x + + x i x x i e S 0 = ts 0bwG, 1 S 0 α j G ts 0bwG. S 0 α j G S 0 ts 0 bwg S 0 + S 0 ts 0 bwg. S 0 the electronic journal of combinatorics , #P4.8 5

6 If n is even, then S 0 = S 0 and bwg 1 n 1 and n j=1 n α j G. If n is odd, then S 0 = S 0 1 = α j G S 0 n + 1bwG bwg =, bwg S 0 n + 1 The isoperimetric number of a -uniform hypergraph G is defined as { } ES, S V G ig = min : S V G, 0 < S. S j=1 n α j G. Let βg = max α j G denote the maximum inverse Perron value of G. In [0], it was j V G shown that ig αg. We improve it as follows. Theorem 8. Let G be a -uniform hypergraph. Then j=1 ig αg + βg. Proof. Suppose that S 1 V G satisfying 0 < S 1 x = x 1,..., x n T be the vector satisfying V G and ES 1,S 1 S 1 = ig. Let x i = Then n x i = 1. For j S 1, we obtain { S1 1, i S1, 0, i S 1. where ts 1 = 1 α j G L G x = ts 1 ES 1, S 1 S 1 e S 1. ES 1,S 1 e ES 1,S 1 Similarly, for j S 1, we get = ts 1 ig, 3 α j G ts 1 ES 1, S 1 S 1 ts 1 ig. 4 Let α s G = βg. If s S 1, by 3, we get From 4, we have βg = α s G ts 1 ig. αg = min α j G min α j G ts 1 ig. j V G j S 1 the electronic journal of combinatorics , #P4.8 6

7 Then αg + βg ts 1 ig + ts 1 ig = ig. Similarly, if s S 1, we can also obtain αg + βg ig. From the above discussion, we get ig αg+βg. The distance du, v between two distinct vertices u and v of G is the length of the shortest path connecting them. The eccentricity of a vertex v is eccv = max{du, v : u V G}. The diameter and radius of G are defined as diamg = max eccv and v V G radg = min eccv, respectively. v V G Theorem 9. Let G be a connected -uniform hypergraph with n vertices. Then eccj 1α j G, j V G. Proof. For j V G, let x = x 1,..., x n T R n + satisfying x j = 0, α j G = L G x. Then α j G = L G x = {i 1,...,i } EG From AM-GM inequality, it yields that n x i = 1 and x + + x i x x i. 5 1 s<t x is x it 1 1 s<t x is x it 1 = 1 x x i. 6 By 5 and 6, we have α j G {i 1,...,i } EG = 1 1 = 1 1 x i x i 1 x x is it {i 1,...,i } EG 1 s<t e EG s,t e 1 s<t x is x it x s x t. 7 Let v 0 {i x i = max p V G x p}. Let P = v 0 e 1 v 1 e v l 1 e l v l be the shortest path from vertex v 0 to vertex v l = j. Then x v 0 1 n 1, x v l = 0 and x s x t x s x t e EG s,t e e EP s,t e the electronic journal of combinatorics , #P4.8 7

8 = = l x vi 1 x vi + l x vi 1 x 1 vi + l u j e i \{v i 1,v i } u j e i \{v i 1,v i } x v i 1 x vi + x vi 1 x vi l x vi 1 x vi. By Cauchy-Schwarz inequality, we obtain e EG s,t e = l x v0 x vl x s x t eccj From 7 and 8, it yields that α j G l x v i 1 x uj + x uj x vi x vi 1 x uj + x uj x vi x vi 1 x vi l x v0 x vl l x vi 1 x vi eccj. 8 1eccj, eccj 1α j G. For a connected -uniform hypergraph G with n vertices, [0] showed that diamg 4 n 1αG. By Theorem 9, we obtain the following improved result. Corollary 10. Let G be a connected -uniform hypergraph with n vertices. Then diamg 1αG, radg 1βG. In [7], it was shown that αg δ, where δ is the minimum degree of G. We improve it as follows. Theorem 11. Let G be a -uniform hypergraph with n vertices. Then α j G 1d j, j V G. the electronic journal of combinatorics , #P4.8 8

9 Proof. For j V G, let x = x 1,..., x n T be the vector satisfying { 1, i j, x i = 0, i = j. Then n x i = 1, and we get α j G L G x = = {i 1,...,i } E j G {i 1,...,i } EG where E j G denotes the set of edges containing j. By Theorem 11, we obtain the following result. x + + x i x x i x + + x i = 1d j, Corollary 1. Let G be a -uniform hypergraph with n vertices and m edges. Then n α j G j=1 1m, j V G. Let G be a -uniform hypergraph. For x, y V G, let cx, y = {e EG : x, y e}. A -n, b,, r, λ design can be regarded as a -uniform r-regular hypergraph G on n vertices, b edges, and cx, y = λ for any pair of distinct x, y V G. A -design satisfying n = b is called a symmetric design. Theorem 13. Let G be a connected -uniform hypergraph with n vertices. Then G is a -design if and only if α 1 G = = α n G = 1, where is the maximum degree of n 1 G. Proof. We first prove the necessity. If G is a -n, b,, r, λ design, then λn 1 = r 1 and = r = d 1 = = d n. For any j V G, by Theorem 11, we have r 1 α j G = λ. 9 n Let x = x 1,..., x n T R n + satisfying x j = 0, x i = 1 and α j G = L G x. Then we get α j G = L G x {i 1,...,i } E j G Combining 9 and 10, we get x + + x i x x i = λ x i = λ. 10 i j α 1 G = = α n G = λ = r 1 = 1. the electronic journal of combinatorics , #P4.8 9

10 Next we prove the sufficiency. Let α 1 G = = α n G = 1. From Theorem n 1 T 11, we obtain d 1 = = d n =. Let z = 1,..., 1 R n 1 +. For j V G, let y = y 1,..., y n T R n + be a vector such that y i = 0 if i = j and y i = 1 otherwise. Then L G y = y + + yi y y i = y + + yi = {i 1,...,i } EG 1 = α j G = αg. {i 1,...,i } E j G We now that αg = α j G is the smallest H-eigenvalue of L G j. Since L G jz = L G y = αg, z is an H-eigenvector corresponding to αg, that is For all i V G \ {j}, we have αgz [ 1] = L G jz 1. αg = 1 LG jz 1 z = 1 1 i i z 1 i = L G ii i = ci, j. i,...,i j i,...i j L G j ii i z i z i So ci, j = αg for any pair of distinct i, j V G, which implies that G is a -design. We give an estimation of the edge connectivity of a -design as follows. Theorem 14. Let G be a -n, b,, r, λ design. Then nλ eg λ 1. Moreover, if G is a symmetric design, then eg = = r. Proof. Since G is a -n, b,, r, λ design, we have λ = r 1. By Theorem 13, we have r 1 αg = = λ. It follows from Lemma that nλ = n λ αg eg r = Moreover, if G is a symmetric design, then n = b. Since nr = b, we have r =. From λ = r 1 and 11, we have eg. Since eg is a positive integer, we get eg = = r. the electronic journal of combinatorics , #P4.8 10

11 4 Inverse Perron values and resistance distance of graphs For a vertex i of a connected graph G, we define its resistance eccentricity as r i G = max j V G r ij. Theorem 15. Let G be a connected graph. For any i V G, we have r i G 1 α i G. Proof. Without loss of generality, assume that i is the vertex corresponding to the last row of the Laplacian matrix L G. Since α i G is the minimum eigenvalue of the principal submatrix L G i, α 1 i G is the spectral radius of the symmetric nonnegative matrix L G i 1. So α 1 i G max L Gi 1 jj. j i By Lemmas 5 and 3, we get r ij G = L G i 1 jj for any j i. Hence α 1 i G max L Gi 1 jj = r i G, j i r i G 1 α i G. For a vertex i of a connected graph G, its resistance centrality is defined as Kf i G = r ij G. It is used to measure the centrality of a networ [4]. Note that KfG = r ij G = 1 Kf i G. j V G {i,j} V G i V G Theorem 16. Let G be a connected graph with n vertices. For any i V G, we have nkf i G KfG α i G. Proof. Note that α 1 i G is the maximum eigenvalue of the symmetric matrix L G i 1. Let e be the all-ones column vector, then By Lemmas 5 and 4, we have α 1 i G e L G i 1 e e e = e L G i 1 e. KfG = ntrl G i 1 e L G i 1 e. From Lemmas 5 and 3, we get r ij G = L G i 1 jj for any j i. Hence trl G i 1 = Kf i G and KfG = nkf i G e L G i 1 e. the electronic journal of combinatorics , #P4.8 11

12 By α 1 i G e L G i 1 e n 1 we get α 1 i G e L G i 1 e nkf i G KfG α i G. = nkf ig KfG, Corollary 17. Let G be a connected graph with n vertices. Then KfG n n α 1 i G. Proof. By Theorem 16, we have n α i G n nkf i G KfG = nkfg, KfG n n α 1 i G. References [1] E. Andrade and G. Dahl. Combinatorial Perron values of trees and bottlenec matrices. Linear Multilinear Algebra, 65: , 017. [] R. B. Bapat. Graphs and Matrices. Springer, London, 010. [3] E. Bendito, A. Carmona, A. M. Encinas and J. M. Gesto. A formula for the Kirchhoff index. Int. J. Quantum Chem., 108: , 008. [4] E. Bozzo and M. Franceschet. Resistance distance, closeness, and betweenness. Social Networs, 35: , 013. [5] A. Bretto. Hypergraph Theory: An Introduction. Springer, New Yor, 013. [6] C. Bu, B. Yan, X. Zhou and J. Zhou. Resistance distance in subdivision-vertex join and subdivision-edge join of graphs. Linear Algebra Appl., 458:454 46, 014. [7] C. Bu, J. Zhou and Y. Wei. E-cospectral hypergraphs and some hypergraphs determined by their spectra. Linear Algebra Appl., 459: , 014. [8] J. Cooper and A. Dutle. Spectra of uniform hypergraphs. Linear Algebra Appl., 436:368 39, 01. [9] C. Cui, Z. Luo, L. Qi and H. Yan. Computing the analytic connectivity of a uniform hypergraph. arxiv: v1, 016. [10] Y. Fan, Y. Tan, X. Peng and A. Liu. Maximizing spectral radii of uniform hypergraphs with few edges. Discuss. Math. Graph Theory, 36: , 016. the electronic journal of combinatorics , #P4.8 1

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