Analytic Connectivity of K-Uniform Hypergraphs

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2 Preliminaries

Let be a real tensor of order m dimension n and x = (x1, x2, , xn) be a vectorT in Cn. Then ···

n xm := a x x , (2.1) T i1,··· ,im i1 ··· im i ,··· ,i =1 1 Xm and xm−1 is a vector in Cn, whose i-th component is deﬁned by T n ( xm−1) = a x x , for i [n]. T i i,i2,··· ,im i2 ··· im ∈ i ,··· ,i =1 2 Xm

2 [r] r r r Cn Let x = (x1, x2,...,xn) be a vector in , where r is a positive integer. If there is a non-zero vector x Cn and λ C, satisfying ∈ ∈ xm−1 = λx[m−1], (2.2) T then one can say λ is an eigenvalue of , and x is its corresponding eigenvector. In particular, if x is real, thenTλ is also real. In this case, λ is an H eigenvalue of . And x is its corresponding H eigenvector[2]. Let Rn− Rn T − + and ++ be the set of all nonnegative vectors and the set of all positive Rn Rn + vectors in , respectively. If x +, then λ is called an H eigenvalue Rn ∈++ − of . If x ++, then λ is an H eigenvalue of . If all the entries are nonnegative,T ∈ then is called a nonnegative− tensor. T In what follows,T we employ standard definitions and notation from hypergraph theory; see, e.g.,[1]. A hypergraph H is a pair (V, E). The elements of V = V (H) = v , v , , v are referred to as vertices and the elements of { 1 2 ··· n} E = E(H) = e1,e2, ,em are called edges, where ei V for i [m]. A hypergraph H {is said to··· be k-uniform} for an integer k 2,⊆ if for all e ∈ E(H), ≥ i ∈ ei = k, where i [m]. We often use the term k-graph in place of “k-uniform hypergraph”| | for∈ short. Clearly, a 2-graph is what is usually termed a simple, undirected graph. For any vertex v V (i [n]), its degree d(v ) is i ∈ ∈ i defined as d(vi) = ep : vi ep E . Denote the maximum degree, the minimum degree and|{ the average∈ degree∈ }| of H by ∆(H), δ(H) and d(H), respectively. If d(H) = ∆(H), then H is a regular k-graph. Two vertices vi and v are called adjacent if and only if there exists an edge e E(H) such that j ∈ vi, vj e E(H). Two vertices vi and vj are called connected if either vi and{ v }are ⊆ adjacent,∈ or there are vertices v , , v such that v and v , v j i1 ··· is i i1 is and vj , vir and vir+1 for r =1, ,s 1 are adjacent, respectively. A k-graph H is called connected if any pair··· of− its vertices are connected. The isoperimetric number for a k-graph H, denoted by i(H), is defined by E (S, S) V (G) i(H) = min | H | : S V (G), 0 S | | , (2.3) S ⊂ ≤ ≤ 2 | | where S = V S and the edge set E (S, S) consist of the edges in H with end \ H vertices in both S and S. Particularly, when k = 2, the edge e EH (S, S) has exactly one vertex in S and one vertex in S. When k 3, the edge∈ e E (S, S) ≥ ∈ H satisfies e S = and e S = . For the sake of convenience, EH (S), or E(S) for short,∩ denotes6 ∅ the edge∩ set6 ∅ consisting of edges in H whose vertices are all in S. Sometimes EH (S, S) is called an edge cut of H. Indeed, if we delete E(S, S) from H, then H is separated into two k-graphs H[S] = (S, E(S)) and H[S] = (S, E(S)). The minimum cardinality of such an edge cut is called the edge connectivity of H, denoted by e(H). A vertex cut of H is a vertex subset V ′ V (H) such that H V ′ is disconnected, where H V ′ is the graph obtained⊂ by deleting all vertices− in V ′ and all incident edges.− The vertex connectivity of H, denoted by v(H) is the minimum cardinality of any vertex cut V ′. The complete k-graph has no vertex cut.

3 For a k-graph H with n vertices, the (normalized) adjacency tensor, denoted by (H) or for short, is a tensor of order k dimension n with entries A A 1 if v , v , , v E(H) a = (k−1)! { i1 i2 ··· ik } ∈ (2.4) i1,i2··· ,ik 0 otherwise. The degree tensor of H, denoted by (H) or for short, is a diagonal tensor D D of order k and dimension n with its i-th diagonal entry as dH (vi) and 0 otherwise. The Laplacian tensor and signless Laplacian tensor of H are deﬁned as = (H)= (H) (H) and = (H)= (H)+ (H), respectively[8]. It Lis easyL to seeD that −, A and areQ allQ symmetric.D Particularly,A when k = 2, the tensors , and A areL theQ adjacency matrix A, the Laplacian matrix L and the signlessA L LaplacianQ matrix Q of a 2-graph, respectively. Let e = v , v , , v E(H). Denote { i1 i2 ··· ik } ∈ k (e)xk = xk kx x x . L ij − i1 i2 ··· ik j=1 X Then xk = (e)xk. L L eX∈E Here we present the arithmetic-geometric mean inequality, which we refer to as the A-G inequality for brevity.

Rn a1+a2+···+an Lemma 2.1. Let a = (a1,a2, ,an) be a vector in + and A(a)= n , 1 ··· G(a) = (a1a2 an) n . then ··· A(a) G(a). ≥ Moreover, the equality holds if and only if a = a = = a . 1 2 ··· n Rn k By A-G inequality, if x +. then (e)x 0 for each edge e E(H), so that xk 0 as well. Moreover,∈ it is easyL to verify≥ that the smallest∈ Laplacian H-eigenvalueL ≥ of H is exactly 0 and the all-ones vector, denoted by 1, is the corresponding eigenvector. In [20], Qi proved that

n 0 = min xk : x Rn , xk =1 . {L ∈ + i } i=1 X Qi also deﬁned the analytic connectivity α(H) of the k-graph H by

n k Rn k α(H) = min min x : x +, xi =1, xj =0 . (2.5) j=1,2,··· ,n {L ∈ } i=1 X Clearly, α(H) 0. The following statements illuminate the name of this parameter. ≥ Theorem 2.2. [20] The k-graph H is connected if and only if α(H) > 0.

4 Theorem 2.3. [20] For a k-graph H, we have n e(G) α(H). ≥ k Recall that the second smallest eigenvalue of Laplacian metric of a 2-graph G of order n, denoted by λ2(G), is deﬁned by Lx, x λ2(G) = min h i , (2.6) x⊥1 xT x x6=0 where 0 is the zero vector. Fiedler obtained another important expression for λ2(G):

2 v v ∈E(G)(xi xj ) λ (G)=2n min i j − : x = c1,c R . (2.7) 2 n n (x x )2 6 ∈ ( Pi=1 j=1 i − j )

Fiedler referred to the numberPλ2(GP) as the algebraic connectivity of G, and related it to the classical connectivity of 2-graphs.

Theorem 2.4. [27] A 2-graph G is connected if and only if λ2(G) > 0. Theorem 2.5. [27] For a 2-graph G, λ (G) v(G) e(G). 2 ≤ ≤ There are several prominent graph classes for which the algebraic connectivity is known. Here we give the algebraic connectivity when G is a complete graph. n Theorem 2.6. If Kn is a complete 2-graph of order n with 2 edges, then

λ2(Kn)= n with corresponding eigenvector x = (n, 1, 1, , 1). − − ··· − There are also several bounds to the algebraic connectivity related to the parameters of a graph, such as degree, isoperimetric number, and diameter.

Theorem 2.7. Let G be a 2-graph with more than one edge and (d(v1), d(v2), , d(v )) be the degree sequence of G. Then ··· n d(vi)+ d(vj ) 2 λ2(G) min − . (2.8) ≤ {vi,vj }∈E(G) 2 In general, the isoperimetric number is very hard to compute, and even obtaining any lower bounds on i(H) seems to be a diﬃcult problem. However, for 2-graphs G, the algebraic connectivity provides a reasonably good bound on i(G). The following is a well-known inequality often called the “Cheeger inequaility”. Theorem 2.8. [24] Let G be a 2-graph. Then

2i(G) α(G) ∆(G) ∆(G)2 i2(G). ≥ ≥ − − p 5 Regarding the diameter of a k-graph H, denoted by diam(H), the following is a lower bound for 2-graphs. Theorem 2.9. [25] Let G be a 2-graph with n vertices. Then 4 λ (G) . 2 ≥ diam(G) n · 3 Main results

Evidently, the analytic connectivity and the algebraic connectivity are closely related to the connectivity of a graph. In this section, we will study the properties of analytic connectivity for k-graphs compared with that of algebraic connectivity for 2-graphs. Given a ﬁnite set X and positive integers k, r, and λ, a 2-design (or bal- anced incomplete block design) is a multiset of k-element subsets of X, called blocks, such that the number of blocks containing any element of X is r and the number of blocks containing any pair of distinct x, y X is λ. If none of the elements of the multiset are repeated, then the 2-design is∈ said to be simple. It is easy to see that a simple 2-design on a set of size n with parameters k, r, and λ as above is the same as a k-uniform hypergraph on n vertices all of whose vertices have degree k, and all of whose codegrees c(x, y)= e E : x, y e for x = y are λ. |{ ∈ ∈ }| 6 Theorem 3.1. Let H be a simple 2-design with n vertices, b blocks, k vertices per block, r blocks containing each vertex, and λ blocks containing each pair of vertices. Further suppose that H has no cut-vertex. Then

α(H)= λ,

1 1 1 1 1 1 with corresponding vector x = ( ) k , ( ) k , , ( ) k , 0 . n−1 n−1 ··· n−1 Proof. It is well known that bk = nr and λ(n 1) = r(k 1). Let x = (x , x , , x ) be a vector satisfying − − 1 2 ··· n 1 1 k ( n−1 ) , if i =1, 2, ,n 1; xi = ··· − 0, if i = n. k Then, for any edge e v1, v2, , vn−1 , we have (e)x = 0. For those edges e containing vertex⊆ {v , we··· have (e})xk = k−1 .L Since there are r edges n L n−1 containing vn, it follows that k 1 α(H) xk = r − = λ. (3.9) ≤ L n 1 − On the other hand, suppose that y = (y ,y , ,y ,y ) Rn is the vector 1 2 ··· n−1 n ∈ + achieving α(H); we may assume without loss of generality that yn = 0. Accord- ing to the A-G inequality, for each edge e E(H), (e)yk 0. In addition, ∈ L ≥

6 for those edges e containing the vertex v , we have (e)yk = k−1 yk , where n L j=1 ij ij = n. Therefore, 6 n−1 P α(H) λ yk = λ. (3.10) ≥ i i=1 X Combining (3.9) and (3.10), we have

α(H)= λ.

It remains to verify that any vector x achieving α(H) has the desired form. If equality holds in (3.10), then every edge e E(H) with vn e satisﬁes k ⊆ 6∈ (e)x = 0, since no edge containing vn contributes to the sum. By the A- GL inequality, each coordinate of x corresponding to a vertex in e has the same value. Since the subgraph induced by V (H) vn is connected, we may conclude n\{−1 k} that x1 = x2 = = xn−1. Moreover, i=1 xi = 1, so that x1 = x2 = = 1 1 ··· ··· x = ( ) k . The result follows. n−1 n−1 P (k) We can now give the analytic connectivity of a complete k-graph Kn , as (k) n−2 follows, since Kn is a 2-design with λ = k−2 .

(k) n−2 Corollary 3.2. α(Kn )= k−2 . Remark 3.1. We compare with Theorem 2.6 by taking k =2 in Theorem 3.1, so that α(Kn)=1. Two additional properties of α(H) are given in Theorem 3.3 and Corollary 3.4. Furthermore, Theorem 3.5 presents an upper bound on α(H) in terms of vertex connectivity.

Theorem 3.3. If H1 and H2 are edge-disjoint hypergraphs with the same vertex set then α(H )+ α(H ) α(H H ). 1 2 ≤ 1 ∪ 2 k k Proof. Since H1 and H2 are edge-disjoint, we have (H1 H2)x = (H1)x + (H )xk. Obviously, α(H H ) α(H )+ α(H ).L ∪ L L 2 1 ∪ 2 ≥ 1 2

Corollary 3.4. If H1 and H2 have the same vertex set and E(H1) E(H2), then α(H ) α(H ). ⊆ 1 ≤ 2 Theorem 3.5. Let H be a hypergraph of order n and v(H) be the vertex connectivity of H. Then

n 2 n v(H) 1 n−v(H) 1 k 1 α(H) − − − 2 − − . ≤ k 2 − k 1 − k 1 n 1 − " − − # − Proof. Let S be a minimum cut set of H, i.e., S = v(H); and let H ,H , ,H | | 1 2 ··· l be the components of H S, with ni = V (Hi) for i [l]. Without loss of − | | ∈n−v(H) generality, suppose n1 = mini∈[l] ni, so that 1 n1 2 and there is a vertex u V (H ). ≤ ≤ ∈ 1

7 Deﬁne a vector x = (x , x , , x ) by 1 2 ··· n 1 1 k if vi = u; xi = n−1 6  0 if v = u.  i n k  ′ It is easy to see i=1 xi = 1. Let H arise from H by adding all edges e containing u such that e S = or e Hi = for all i 2,...,l ; note that all edges of H have this form,P∩ so6 H∅ H′∩. The maximum∅ possible∈{ number} of edges contain- n−1 ⊂ ′ ing u is k−1 . However, we exclude from H those edges e such that e S = l n−1 n−v(H)−1 ∩n −1 ∅ and e ( V (H )) = . Therefore, d ′ (u) = + 1 . ∩ j=2 j 6 ∅ H k−1 − k−1 k−1 Moreover, (e)xk = k−1 if e contains vertex u and (e)xk = 0 otherwise. Then, S n−1 by CorollaryL 3.4, we have L

α(H) α(H′) (H′)yk ≤ ≤L n 1 n v(H) 1 n 1 k 1 = − − − + 1 − − k 1 − k 1 k 1 n 1 − − − − n 2 n v(H) 1 n−v(H) 1 k 1 − − − 2 − − ≤ k 2 − k 1 − k 1 n 1 − " − − # − The result follows.

Remark 3.2. Taking k =2, i.e., for a 2-graph G, we have α(G) n+v(G)−2 . ≤ 2(n−1) Next, we investigate upper and lower bounds on α(H) in terms of the isoperimetric number and diameter. Before coming to our results, two extended A-G inequalities are needed.

Rn a1+a2+···+an Lemma 3.6. Let a = (a1,a2, ,an) be a vector in + and A(a)= n , 1 ··· G(a) = (a a a ) n . 1 2 ··· n

(1) [26] Suppose that bj = aσ(i), where j = 1, 2, ,n and σ Sn is a permutation of the set [n], then ··· ∈

m 1 A(a) G(a) ( b b )2, (3.11) − ≥ n j − n+1−j j=1 X p p n where m = 2 . Moreover, equality holds if and only if b1bn = bj bn+1−j , 1 j m. ⌊ ⌋ ≤ ≤ (2) 1 2 A(a) G(a) (√ai √aj ) . (3.12) − ≥ (n 1)n − − 1≤Xi

8 Proof. Here, it is suﬃcient to verify (3.12). Since 1 A(a)= (b + b ), (n 1)n i j − 1≤Xi

α(H) yk = + + (e )yk ≤ L   L p ep∈XE(S) ep∈XE(S) ep∈XE(S,S)  k  = xi v ∈e ∩S ep∈XE(S,S) i Xp 1 t(S) E(S, S) ≤ S | | | | =t(S)i(H). (3.13)

9 Similarly, we have α(H) t(S)i(H). (3.14) ≤ Summing (3.13) and (3.14), since t(S)+ t(S)= k, we have

k α(H) i(H). ≤ 2 Thus, the proofs for the Cheeger inequality regarding an upper bound for α(H) is completed. On the other hand, to verify the lower bound of α(H) for a bipartition (S, S¯) on V (H), we ﬁrst deﬁne a multiple 2-graph with possible loops, H = (V (H), E(H)), where V (H) = V (H) and the edge set of H is derived in the following way. Suppose x = x , x , , x is the vector achieving α(H).b For { 1 2 ··· n} eachb edge eb= v , v , b, v E(H), with loss of generality,b let x x { i1 i2 ··· ik } ∈ i1 ≥ i2 ≥ x . Then E(H)= v v : j =1, 2, , k . ···≥ ik e∈E(H){ ij k+1−ij ··· ⌊ 2 ⌋} Since x 0 for i [k], then i S ≥ b ∈ n α(H)= xk kx x  ij − i1 ··· ik  e={v ,··· ,v }∈E j=1 i1 X ik X k  2 xk xk (By (3.11)) ≥ ij − k+1−ij e={v ,··· ,v }∈E j=1 i1 X ik X q q xi1 ≥xi2 ≥···≥xik 2 = xk xk i − j vi,vjX∈E(Hb) q q

k/2 k/2 2 [ k ] Let M = (xi xj ) and y = x 2 . Moreover, for the sake of (vi,vj )∈E(Hb) − ′ convenience,P we denote by E the set E(H). According to the proof of Theorem 2.8, we have b

2 2 ′ (yi yj ) ′ (yi + yj) vi,vj ∈E − vi,vj ∈E M = 2 (3.15) ′ (y + y ) P vi,vj ∈E Pi j 2 2 2 ( ′ y y ) vi,vj ∈EP| i − j | 2 (by Cauchy-Schwarz) ≥ ′ (yi + yj ) Pvi,vj ∈E P Let 0 = t0 < t1 < t2 < < th be all distinct values of yi, i = 1, 2, ,n. For s =0, 1, 2, ,h, let V ···= v V : y t . For each edge e E (V···, V ), ··· s { i ∈ i ≥ s} ∈ H s s let δs(e) = min Vs e , Vs e . Denote δ(Vs) = min δs(e): e EH (Vs, Vs) and δ(H) = min{| ∩ δ(|V| ) .∩ Then|} { ∈ } s∈[h]{ s }

10 h y2 y2 = (y2 y2) | i − j | i − j ′ i=1 ′ vi,vXj ∈E X vivXj ∈E yi≥yj h = (t2 t2 + t2 t2 + t2 t2) r − r−1 r−1 −···− l+1 l+1 − l i=1 yi=tr X yj =Xtl,l

On the other hand,

2 2 2 2 (yi + yj ) =2 (yi + yj ) (yi yj ) ′ ′ − ′ − vivXj ∈E vivXj ∈E vivXj ∈E n 2 d y2 (y y )2 ≤ i i − i − j i=1 ′ X vivXj ∈E n 2∆(H) y2 (y y )2 ≤ i − i − j (3.17) i=1 ′ X vivXj ∈E b n =(2∆(H) M) y2 − i i=1 X b n (2∆(H) M) y2 ≤ − i i=1 X where ∆ denotes maximum degree. Combining (3.15), (3.16) and (3.17), we obtain δ(H)2i(H)2 M . ≥ 2∆(H) M −

11 Therefore, M ∆(H) ∆(H)2 δ(H)2i(H)2. ≥ − − Since δ(H) 1, we have α(H) ∆(H) ∆(H)2 i(H)2. ≥ ≥ p − − Theorem 3.8. Let H be a k-graph. Thenp 4 α(H) . (3.18) ≥ n2(k 1) diam(H) − Proof. Let x = (x1, x2, , xn) be the vector achieving α(H), where xn = 0, [ k ] ··· ∗ and y = x 2 . Deﬁne a multiple 2-graph H as follows. It has vertex set V (H), and vertices u and v are adjacent in H∗ if and only if u, v e E(H). Evidently, diam(H) = diam(H∗). Therefore, { } ⊂ ∈

α(H)= xk = (e)xk L L e∈XE(H) k = xk kx x x  ij − i1 i2 ··· ik  e={v ,v ,··· ,v }∈E(H) j=1 i1 i2 X ik X  1  2 xk/2 xk/2 (by (3.12)) ≥ k 1 is − it e={v ,v ,··· ,v }∈E(H) 1≤s

12 Hence, from Theroem 2.9,

λ (H∗) 4 4 α(H) 2 = . ≥ n(k 1) ≥ n2(k 1) diam(H∗) n2(k 1) diam(H) − − − Completing the proof. The last theorem gives an upper bound on the analytic connectivity of k- graphs as a function of degree sequence. Theorem 3.9. Let H be a k-graph with more than one edge. Then

d(v )+ d(v )+ + d(v ) k α(H) min i1 i2 ··· ik − : v , v , , v E(H) . ≤ k { i1 i2 ··· ik } ∈ (3.20)

Proof. Let ep = vi1 , vi2 , , vik be an edge in H and x = (x1, x2, , xn) be a vector deﬁned{ by ··· } ···

k−1/k if v e , x = i p i 0 otherwise∈ . Then n xk = 1. Moreover, it is easy to see that (e )xk = 0. Then, i=1 i L p αP(H) xk = (e)xk ≤L L e∈XE(H) =(d(v ) 1)(1/k) + (d(v ) 1)(1/k)+ + (d(v ) 1)(1/k) i1 − i2 − ··· ik − d(v )+ d(v )+ + d(v ) k = i1 i2 ··· ik − k completing the proof.

Remark 3.3. When k =2, the upper bound in inequality (3.20) is

d(v )+ d(v ) 2 min i1 i2 − : v , v E(H) , 2 { i1 i2 } ∈ which is exactly the upper bound in (2.8).

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