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ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS ASHWIN GUHA, MUNI SREENIVAS PYDI, BISWAJIT PARIA, AND AMBEDKAR DUKKIPATI Abstract. Inthis paper we extendthe knownresultsof analytic 7 1 connectivity to non-uniform hypergraphs. We prove a modified 0 Cheeger’sinequalityandalsogiveaboundonanalyticconnectivity 2 withrespecttothedegreesequenceanddiameterofahypergraph. n a J 7 1 1. Introduction ] M Combinatorial graph theory is a vast active area of research with a D wide variety of applications. It is well-known that many interesting . s problems have no polynomial-time algorithms. In an attempt to find c approximate solutions to these problems, matrices pertaining to the [ graph such as adjacency matrix or Laplacian matrix were studied, and 1 v the eigenvalues of these matrices were used to bound various graph 8 parameters. This gave rise to spectral graph theory, which has since 4 become a separate area of research in its own right. 5 4 The idea of graphs has been generalized to hypergraphs where an 0 edge may span more than two vertices. Hypergraphs have also been . 1 explored in detail [2]. However, spectral methods for hypergraphs have 0 received less attention. Recently there has been a renewed interest in 7 1 spectral hypergraph theory. : v Thetraditional approachtowardsdealing with hypergraphs hasbeen i to convert it into a graph and apply the known graph results. Another X approach is to define a tensor for a hypergraph that is an analogue of r a the corresponding matrix for a graph. A novel definition of eigenvalue for a tensor given by Qi [8] and independently by Lim [7] has led to surge of activity in this area. Subsequently various results on spectra of hypergraphs have been provided in [9, 4] . The results in these works are applicable to uniform hypergraphs where the size of every edge is fixed. There have been recent attempts to extend these results for non-uniform hypergraphs [1, 3]. The concept of analytic connectivity was introduced in [9], which is the analogue of algebraic connectivity introduced by Fiedler [5] for graphs. In [6], the relations between analytic connectivity and other graph parameters are explored. In particular, Cheeger’s inequality for uniform hypergraphs is proved. 1 2 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS In this paper we use the terminology given by Banerjee et al. [1] and extend the results by Li et al. [6] to non-uniform hypergraphs. We prove a modified Cheeger’s inequality and also give the relation between analytic connectivity and diameter. The paper is organized as follows. In Section 2 we introduce our notation and state the preliminaries. We state the equivalent theorems for graphs and uniform hypergraphs. In Section 3, we provide the main results with proof. Section 4 contains concluding remarks. 2. Preliminaries Let H(V,E) be a general hypergraph on vertex set V and edge set E which is a set of non-empty subsets of V. Let n be number of vertices. We denote by m and s the size of the largest edge and the smallest min edge respectively, i.e., m = max e : e E and s = min e : e min {| | ∈ } {| | ∈ E . Let ∆ be the largest degree among vertices of H. } For a hypergraph H we adopt the definition given in [1] and define the adjacency tensor to be an n-dimensional hypermatrix 1 of order A msuchthatife = v ,v ,...,v E isanedgeofcardinalitys m, { i1 i2 is} ∈ ≤ then s m! a = , where Ω = p1p2...pm Ω k !k !...k ! 1 2 s kXj≥1 with k = m and p ,p ,...,p are chosen in all possible ways from j 1 2 m i ,i ,...,i such that each element of the set appears at least once. 1 2 s { P } The other entries of the hypermatrix are zero. Note that this definition agrees with the definition of adjacency tensor for uniform hypergraphs given in [4]. For an edge e = v ,v ,...,v E we define { i1 i2 is} ∈ xe = x x ...x , m r1 r2 rm X where the sum is over r ,r ,...,r chosen in all possible ways from 1 2 m i ,i ,...,i with each element of the set appearing at least once in 1 2 s { } the index . Let be the degree tensor which is a diagonal tensor of order m D and dimension n such that d = d(v ) and zero elsewhere. We define ii...i i Laplacian tensor (H) to be . For a tensor Lof order mDan−dAdimension n and x Rn we define T ∈ xm as T n ( xm) = t x x ...x . T j ji2i3...im i2 i3 im i2,.X..,im=1 1Inthis paperweuse the termshypermatrixandtensorinterchangeablyforeaseof understanding. ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS 3 In particular we have n n xm = ( )xm = d(v )xm a x ...x L D −A i i − ii2...im i2 im Xi=1 i2,.X..,im=1 s s = xm xe = (e)xm ij − Ω m L ! e={vi1X,...,vis}∈E Xj=1 Xe∈E s s where (e)xm = xm xe . L ij − Ω m j=1 It can be shownXthat for x Rn, (e)xm 0 [1]. We recall the ∈ + L ≥ definitions of a few hypergraph parameters which are intuitive gener- alizations of those for a 2-graph. Definition 2.1. For a hypergraph H, the isoperimetric number i(H) is defined as ∂S V i(H) = min | | : S V,0 S | | , S ⊂ ≤ | | ≤ 2 (cid:26) | | (cid:27) where ∂S is the boundary of S which consists of the edges in H with vertices in both S and S = V S. When H is a 2-graph, the edges in \ ∂S have exactly one vertex in S and one vertex in S. Definition 2.2. For a hypergraph H, the diameter diam(H) is defined as the maximum distance between any pair of vertices. diam(H) = max l(u,v) : u,v V , { ∈ } where l(u,v) is the length of the shortest path connecting u and v. The concept of algebraic connectivity, defined as the second smallest eigenvalue of the Laplacian (denoted by λ ) was introduced by Fiedler 2 [5]. The algebraic connectivity has proved to be a reliable measure to understand the structure of a 2-graph. Some of the results which give bounds on λ with respect to various graph constants are given below, 2 including a version of the famous Cheeger inequality. Lemma 2.3. For a graph G with n vertices, (x x )2 λ (G) = 2nmin (vi,vj)∈E i − j : x = c.1 , for all c R . 2 n n (x x )2 6 n ∈ ( Pi=1 j=1 i − j ) Theorem 2.4. ForPa graPph G with n vertices, 4 λ (G) . 2 ≥ n diam(G) · Theorem 2.5. LetG be a 2-graphwith more thanone edgeand(d(v ),...,d(v )) 1 n be the degree sequence of G. Then d(v )+d(v ) 2 i j λ (G) min − . 2 ≤ {vi,vj}∈E(cid:26) 2 (cid:27) 4 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS Theorem 2.6. For a graph G, 2i(G) λ (G) ∆(G) ∆(G)2 i(G)2. 2 ≥ ≥ − − For k-graphs the concept of analytic cponnectivity was introduced by Qi [9] as the equivalent of λ . The definition is also valid for non- 2 uniform hypergraphs as mentioned in [1]. Definition 2.7. The analytic connectivity of a k-uniform hypergraph H is defined as n α(H) = min min xm : x Rn, xm = 1,x = 0 . j=1,...,n {L ∈ + i j } i=1 X The above results for 2-graphs have been extended to k-graphs in [6] where the inequalities are presented with respect to α(H). The theorems are given below. Theorem 2.8. Let H be a k-graph. Then 4 α(H) . ≥ n2(k 1)diam(H) − Theorem 2.9. Let H be 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 . ≤ k { i1 i2 ik} ∈ (cid:26) (cid:27) Theorem 2.10. For a k-graph H with k 3, ≥ k i α(H) ∆ √∆2 i2. 2 ≥ ≥ − − We also mention a lemma from [6] which will be used to prove the main results in the next section. Lemma 2.11. Let a = (a ,...,a ) Rn. Let A = (a +...+ a )/n 1 n ∈ + 1 n and G = (a ...a )1/n. Then 1 n 1 A G (√a √a )2. (1) i j − ≥ n(n 1) − − 1≤i<j≤n X ⌊n/2⌋ 1 A G ( b b )2. (2) j n+1−j − ≥ n − j=1 X p p where b = a , for j = 1,...,n and σ is a permutation of the set j σ(j) 1,...,n . { } 3. Results for general hypergraphs Inthissectionweprovethecorrespondingresultsfornon-uniformhy- pergraphs. The inequalities are similar to that of uniform hypergraphs, except for an additional factor of smin. The proofs can be obtained by m modifying those in [6]. We have given the detailed proofs here for the sake of clarity and completeness. ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS 5 Theorem 3.1. Let H be a general hypergraph. Then 4smin α(H) . ≥ n2m(m 1)diam(H) − Proof. Let x = (x ,x ,...,x ) be the vector achieving α(H). Assume 1 2 n x = 0 and let y = x(m/2). Define a 2-graph H∗ with vertex set V(H) n and u v in H∗ if and only if u,v e E. In other words H∗ is ∼ { } ⊂ ∈ the clique expansion of H. We know that diam(H) = diam(H∗). For edge e = v ,...,v E , consider k copies of xm, k copies of xm { i1 is} ∈ 1 1 2 2 and k copies of xm, where k 1 and k = m. From Lemma 2.11 s s j ≥ j we have, P k xm +k xm +...k xm 1 1 2 2 s s xk1xk2 ...xks m − 1 2 s 1 [(xm/2 xm/2)2 +(xm/2 xm/2)2 +...+(xm/2 xm/2)2] ≥ m(m 1) 1 − 2 1 − 3 s−1 − s − 1 = (xm/2 xm/2)2. m(m 1) i − j − 1≤i<j≤s X Summing over different values of k ,k ,...,k we get 1 2 s s Ω Ω xm xe (xm/2 xm/2)2 s i − m ≥ m(m 1) i − j ! i=1 − 1≤i<j≤s X X s 1 s 1 xm xe (xm/2 xm/2)2 s i − Ω m ≥ m(m 1) i − j ! i=1 − 1≤i<j≤s X X s 1 (e)xm (xm/2 xm/2)2 L ≥ m(m 1) i − j − 1≤i<j≤s X s 1 min (xm/2 xm/2)2. ≥ m (m 1) i − j − 1≤i<j≤s X α = (e)xm L e∈E(H) X s 1 min (xm/2 xm/2)2 ≥ m (m 1) i − j − (vivjX)∈E(H∗) s 1 = min (y y )2 i j m (m 1) − − (vivjX)∈E(H∗) = smin 1 n n (y y )2 (vivj)∈E(H∗)(yi −yj)2 m (m 1) i − j n n (y y )2 − i=1 j=1 P i=1 j=1 i − j XX λ (H∗) s n n P P 2 min (y y )2 from Lemma 2.3. i j ≥ 2n(m 1) m − − i=1 j=1 XX 6 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS n n n n n n n n (y y )2 = y2 + y2 2 y y i − j i j − i j i=1 j=1 i=1 j=1 i=1 j=1 i=1 j=1 XX XX XX XX 2 n n−1 = 2n y2 2 y (since y = 0) i − i n ! ! i=1 i=1 X X n 2n 2(n 1)( y2) (from Cauchy-Schwarz) ≥ − − i i=1 X = 2. We have λ (H∗) s 2 min α 2. ≥ 2n(m 1) m · − From Theorem 2.4, λ (H∗) 4 . Therefore, 2 ≥ diam(H∗)·n s 4 min α . ≥ m n2(m 1) diam(H∗) − · (cid:3) Theorem 3.2. Let H be a non-uniform hypergraph with more than one edge. Then d(v )+d(v )+...+d(v ) s α(H) min i1 i2 is − : v ,v ,...,v E . ≤ s { i1 i2 is} ∈ (cid:26) (cid:27) Proof. Let e = v ,v ,...,v E. Define a vector x Rn such 0 { i1 i2 is} ∈ ∈ + that s−m if v e i 0 x = ∈ i 0 otherwise. ( m m 1 s Then xm = s( )m = 1. (e )xm = xm xe = 0. i s1/m L 0 i − Ω m i=1 i=1 X X α(H) xm ≤ L = (e)xm L e∈E X = (e)xm + (e )xm 0  L L  e∈EX\{e0} = (d(v ) 1)(1/s)+(d(v ) 1)(1/s)+...+(d(v ) 1)(1/s) i1 − i2 − is − d(v )+d(v )+...+d(v ) s = i1 i2 is − . s (cid:3) Theorem 3.3. For a non-uniform hypergraph H, m smin i α(H) (∆ √∆2 i2). 2 ≥ ≥ m − − ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS 7 Proof. Suppose S V gives the isoperimetric number i. Let y = (y ,...,y ) Rn b⊂e the vector defined as follows. 1 n ∈ + 1 if v S y = |S|1/m i ∈ i (0 otherwise. Let t (e) = v : v e S be the number of vertices of e in S and S |{ ∈ ∩ }| t(S) = Pe∈∂StS(e). |∂S| t (e)+t (e) t(S)+t(S) = e∈∂S S S m. (3) ∂S ≤ P | | α ym = + + (e)ym. (4) ≤ L  L Xe∈S Xe∈S eX∈∂S   Ife = v ,...,v S, (e)ym = ym sye = 0. Ife = v ,...,v { 1 s} ⊂ L i −Ω m { 1 s} ⊂ S, P m s (e)ym = ym ye L i − Ω m i=1 X s s 1 = 1 S − Ω S · | | | | = 0. X Therefore only edges in the boundary contribute to the sum of (4). α ym ≤ i eX∈∂SviX∈e∩S t (e) 1 S = = t(S) ∂S S S | | e∈∂S | | | | X = t(S) i. · Similarly we can get α t(S) i. Adding them we get 2α (t(S) + ≤ · ≤ t(S))i. Combining with (3), we get α mi. ≤ 2 To prove the lower bound, suppose x = (x ,...,x ) achieves α. 1 n For each edge e = v ,v ,...,v assume x x ... x by { i1 i2 is} i1 ≤ i2 ≤ ≤ is rearranging the vertices. We define a 2-graph H whose vertex set is same as H and edges are such that E(H) = v v : j = ∪e∈E(H){ ij is+1−j 1,..., s/2 . Then b ⌊ ⌋} b s s α = ( xm xk1 ...xks). (5) ij − Ω i1 is e={vi1X,vi2,...,vis} Xj=1 kXi≥1 Pki=m 8 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS Consider k copies of xm, k copies of xm and k copies of xm, where 1 1 2 2 s s k 1 and k = m. Applying Lemma 2.11 we get j j ≥ P ⌊m/2⌋ k xm +k xm +...k xm 1 1 1 2 2 s s xk1xk2 ...xks ( b b )2 m − 1 2 s ≥ m i − m+1−i i=1 X p p (6) where b ,...,b is any permutation of the variables x . In particular 1 m i considertheassignmentb = xm,b = xm,...,b = xm ,b = 1 i1 2 i2 ⌊s/2⌋ is/2 m+1−⌊s/2⌋ xm ,...,b = xm, and b ,...,b are assigned to any is+1−⌊s/2⌋ m is ⌊s/2⌋+1 m−⌊s/2⌋ of the remaining m s variables. Then − ⌊m/2⌋ ⌊s/2⌋ ( b b )2 ( b b )2 ( since s m) i m+1−i i m+1−i − ≥ − ≤ i=1 i=1 X p p X p p ⌊s/2⌋ = ( xm xm )2 ij − is+1−j j=1 X q q Using the above inequality with (6), and summing over all possible values of k ,...,k as in proof of Theorem 3.1 we get 1 s ⌊s/2⌋ s α ( ( xm xm )2) ≥ m ij − is+1−j e={vi1,..X.,vis}∈E(H) Xj=1 q q ⌊s/2⌋ s min ( ( xm xm )2) ≥ m ij − is+1−j e={vi1,..X.,vis}∈E(H) Xj=1 q q s = min ( xm xm)2). m i − j {vi,vXj}∈E(Hb) p q Proceeding as in proof of k-graphs in [6], let M = (y y )2 {vi,vj}∈E′ i− j where E′ = E(H) and y = √xm. From Cauchy-Schwarz inequality we i i P have b ( y2 y2 )2 M {vi,vj}∈E′| i − j| . (7) ≥ (y +y )2 P{vi,vj}∈E′ i j P Let w (= 0) < w < ... < w be the distinct values of y , for 0 1 h i i = 1,...,n. For j = 0,...,h, let V = v V : y w . For j i i j { ∈ ≥ } each edge e ∂(V ), let δ (e) = min V e , V e . Let δ(V ) = j j j j j ∈ {| ∩ | | ∩ |} min δ (e) : e ∂(V ) and δ(H) = min δ(V ). j j j=0,...,h j { ∈ } ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS 9 h h y2 y2 = (y2 y2) = (w2 w2) | i − j| i − j k − l {vi,Xvj}∈E′ Xk=1{vi,Xvj}∈E′ Xk=1{vi,Xvj}∈E′ vvji∈∈/VVkk yyijl=<=wkwkl h = (w2 w2 )+(w2 w2 )+...+(w2 w2) k − k−1 k−1− k−2 l+1 − l Xk=1{vi,Xvj}∈E′ yi=wk yj=wl l<k h = (w2 w2 ) k − k−1 Xk=1{vi,Xvj}∈E′vXj∈/Vk vi∈Vk h δ(V ) ∂V (w2 w2 ) ≥ k | k| k − k−1 k=1 X h δ(H)i(H) V (w2 w2 ) ≥ | k| k − k−1 k=1 X = δ(H)i(H)( V (w2 w2 )+...+ V (w2 w2)) | h| h − h−1 | 1| 1 − 0 = δ(H)i(H)(( V V )w2 +...+( V V )w2) | h|−| h−1| h | 1|−| 2| 1 n = δ(H)i(H) y2 i i=1 X (y +y )2 = 2 (y2+y2) (y y )2 i j i j − i − j {vi,Xvj}∈E′ vi,Xvj∈E′ vi,Xvj∈E′ n 2 d(v )y2 (y y )2 ≤ i i − i − j Xi=1 vi,Xvj∈E′ n 2∆(H) y2 (y y )2 ≤ i − i − j Xi=1 vi,Xvj∈E′ b = 2∆(H) M 2∆(H) M. − ≤ − b δ(H)2i(H)2 i(H)2 M , ≥ 2∆ M ≥ 2∆ M − − solving which we get M ∆ √∆2 i2. Substituting in (7) we get ≥ − − s α min(∆ √∆2 i2). ≥ m − − (cid:3) 10 ANALYTIC CONNECTIVITY IN GENERAL HYPERGRAPHS 4. Summary In this paper we have built upon the results for analytic connectivity of k-graphs given in [6] by applying the definitions for general hypergraphs found in [1]. We have obtained bounds for the analytic connectivity of general hypergraphs with respect to diameter and degree sequence of the hypergraph. We also proved a version of Cheeger’s inequality for hypergraphs. References [1] Anirban Banerjee, Arnab Char, and Bibhash Mondal. Spectra of general hypergraphs. arXiv preprint:1601.02136, 2016. [2] ClaudeBergeandEdwardMinieka.Graphs and hypergraphs, volume7.North- Holland publishing company Amsterdam, 1973. [3] Changjiang Bu, Jiang Zhou, and Lizhu Sun. Spectral properties of general hypergraphs. arXiv preprint:1605.05942, 2016. [4] Joshua Cooper and Aaron Dutle. Spectra of uniform hypergraphs. Linear Al- gebra and its Applications, 436(9):3268–3292,2012. [5] MiroslavFiedler.Algebraicconnectivityofgraphs.Czechoslovak mathematical journal, 23(2):298–305,1973. [6] Wei Li, Joshua Cooper, and An Chang. Analytic connectivity of k-uniform hypergraphs. Linear and Multilinear Algebra, pages 1–13, 2016. [7] L.H. Lim. Singular values and eigenvalues of tensors, a variational approach. Proceedings of 1st IEEE international workshop on computational advances of multitensor adaptive processing, pages 129–132,2005. [8] Liqun Qi. Eigenvalues of a real supersymmetric tensor. Journal of Symbolic Computation, 40(6):1302–1324,2005. [9] Liqun Qi. H+ eigenvalues of laplacian and signless laplacian tensor. Commu- nications in Mathematical Sciences, 12(6):1045–1064,2014. Department of Computer Science and Automation, Indian Institute of Science, Bangalore 560012, India. E-mail address: [email protected]