Inverse Perron values and connectivity of a uniform hypergraph

In this paper, we show that a uniform hypergraph $\mathcal{G}$ 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 $\mathcal{G}$ in terms of inverse Perron values. By using the inverse Perron values, we give an estimation of the edge connectivity of a $2$-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.


Introduction
Let V (G) and E(G) denote the vertex set and edge set of a hypergraph G, respectively. G is k-uniform if |e| = k for each e ∈ E(G). In particular, 2-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)| denotes the degree of i. The adjacency tensor [8] of a k-uniform hypergraph G, denoted by A G , is an order k dimension |V (G)| tensor with entries The Laplacian tensor [27] 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][8][9][10]14,19,24]. The spectral properties of the Laplacian tensor of hypergraphs are studied in [13,25,27,29,35]. The algebraic connectivity of a graph plays important roles in spectral graph theory [11]. Analogue to the algebraic connectivity of a graph, Qi [27] defined the analytic connectivity of a k-uniform hypergraph G as α(G) = min j=1,...,n min L G x k : x ∈ R n + , 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 α(G) > 0. In [20], some bounds on α(G) were presented in terms of degree, vertex connectivity, diameter and isoperimetric number. A feasible trust region algorithm of α(G) was give in [9].
For any vertex j of uniform hypergraph G, we define the inverse Perron value of j as Clearly, the analytic connectivity α(G) = min 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 a nonsingular M-matrix, 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. The Perron values have close relations with the Fielder vector of a tree [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 Kf (G), is defined as the sum of resistance distances between all pairs of vertices in G, i.e., Kf (G) = {i,j}⊆V (G) r ij (G). Kf (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][5][6]12,26,35,40]. This paper is organized as follows. In Section 2, 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 [20,27]. We also use the inverse Perron values to estimate the edge connectivity of 2-designs. In Section 4, some inequalities among the inverse Perron values, resistance distance and Kirchhoff index of a connected graph are presented. . , x n ) T ∈ R n , let T x m−1 ∈ R n denote the vector whose i-th component is

Preliminaries
and let T x m denote the following polynomial In 2005, Qi [26] and Lim [21] proposed the concept of eigenvalues of tensors, independently. For T = (t i 1 i 2 ···im ) ∈ R [m,n] , if there exists 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, where we have the following lemma.
A path P of a uniform hypergraph G is an alternating sequence of vertices and edges v 0 e 1 v 1 e 2 · · · 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 [2]. Let G be a k-uniform hypergraph, S be a proper nonempty subset of V (G). Denote S = V (G) \ S. The edge-cut set E(S, 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 e(G).
Let tr(A) denote the trace of the square matrix A, and let e denote an all-ones column vector.
Lemma 2.5. [30] Let G be a connected graph of order n. Then Lemma 2.6. [36] Let G be a connected graph of order n. Then where i is the vertex corresponding to the last row of L G .

Inverse Perron values of uniform hypergraphs
In the following theorem, the relationship between inverse Perron values and connectivity of a hypergraph is presented.
Theorem 3.1. Let G be a k-uniform hypergraph. Then the following are equivalent: Proof. (1)=⇒ (2). If G is connected, then by Lemma 2.2, we know that α j (G) > 0 for all j ∈ V (G).
Clearly, we have The bipartition width of a hypergraph G is defined as [18,28] bw where n 2 denotes the maximum integer not larger than n 2 . The computation of bw(G) is very difficult even for the graph case. In [22], Mohar and Poljak use the algebraic connectivity to obtain a lower bound on the bipartition width of a graph. In the following, we use the inverse Perron values to obtain a lower bound on the bipartition width of a uniform hypergraph. Proof. Suppose that S 0 ⊆ V (G) satisfying |S 0 | = n 2 and |E(S 0 , S 0 )| = bw(G). Let x = (x 1 , . . . , x n ) T be the vector satisfying Then n i=1 x k i = 1. For j ∈ S 0 , we can obtain Similarly, for j ∈ S 0 , we can obtain Combining (3.1) and (3.2), we can get The isoperimetric number of a k-uniform hypergraph G is defined as Let β(G) = max j∈V (G) α j (G) denote the maximum inverse Perron value of G. In [20], it is shown that i(G) ≥ 2 k α(G). We improve it as follows. . , x n ) T be the vector satisfying Then n i=1 x k i = 1. For j ∈ S 1 , we can obtain where t(S 1 ) = 1 |E(S1,S1)| e∈E(S 1 ,S 1 ) |e ∩ S 1 |.
For a connected k-uniform hypergraph G with n vertices, [20] showed that By Theorem 3.4, we obtain the following improved result.
Corollary 3.5. Let G be a connected k-uniform hypergraph with n vertices. Then .
In [27], it is shown that α(G) ≤ δ, where δ is the minimum degree of G. We improve it as follows.
Theorem 3.6. Let G be a k-uniform hypergraph with n vertices. Then Proof. For j ∈ V (G), let x = (x 1 , . . . , x n ) T be the vector satisfying Then n i=1 x k i = 1, and we can get By Theorem 3.6, we obtain the following result.
Corollary 3.7. Let G be a k-uniform hypergraph with n vertices, m edges. Then A 2-(n, b, k, r, λ) design can be regarded as a k-uniform r-regular hypergraph G on n vertices, b edges, and c(x, y) = |{e ∈ E(G) : x, y ∈ e}| = λ for any pair of distinct x, y ∈ V (G). A 2-design satisfying n = b is called a symmetric design.
Theorem 3.8. Let G be a connected k-uniform hypergraph with n vertices. Then G is a 2-design if and only if α 1 (G) = · · · = α n (G) = ∆(k−1) n−1 , where ∆ is the maximum degree of G.
Similarly, we can obtain which implies that G is a 2-design.
we give an estimation of the edge connectivity of a 2-design as follows.
Theorem 3.9. Let G be a 2-(n, b, k, r, λ) design. Then Moreover, if G is a symmetric design, then e(G) = k = r.
Proof. Since G is a 2-(n, b, k, r, λ) design, we have λ(n − 1) = r(k − 1). By Theorem 3.8, we have It follows from Lemma 2.3 that Moreover, if G is a symmetric design, then n = b. Since nr = bk, we have r = k.

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 4.1. Let G be a connected graph. For any i ∈ V (G), we have 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 nonneg- By Lemma 2.6, N = L G (i) −1 0 0 0 ∈ R n×n is a symmetric {1}-inverse of L G .
From Lemma 2.4, we get r ij (G) = (L G (i) −1 ) jj for any j = i. Hence .
For a vertex i of a connected graph G, its resistance centrality is defined as Kf i (G) = j∈V (G) r ij (G). It is a centrality index of networks [5]. Proof. Without loss of generality, assume that i is the vertex corresponding to the last row of the Laplacian matrix L G . Then 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