On the critical group of the n-cube

Linear Algebra and its Applications 369 003 51 61 wwwelseviercom/locate/laa On the critical group of the n-cube Hua Bai School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Received 1 May 00; accepted 10 December 00 Submitted by R Guralnick Abstract Reiner proposed two conjectures about the structure of the critical group of the n-cube Q n In this paper we confirm them Furthermore we describe its p-primary structure for all odd primes p The results are generalized to Cartesian product of complete graphs K n1 K nk by Jacobson, Niedermaier and Reiner 003 Published by Elsevier Science Inc Keywords: n-cube; Critical group; Sandpile group; Laplacian matrix; Smith normal form; Sylow p-group 1 Introduction The n-cube or hypercube Q n is the simple graph whose vertices are the n-tuples with entries in {0, 1} and whose edges are the pairs of n-tuples that differs in exactly one position Let G V, E be a finite graph without self-loops, but, with multiple edges allowed and let n V Then n Laplacian matrix LG for this graph G, is defined by { degg u if u v, LG u,v u,v otherwise, where m u,v denotes the multiplicity of the edge {u, v} in E The Laplacian matrix of a graph, which dates back to Kirchhoff s matrix-tree theorem, plays an important role in the study of spanning trees, graph spectra, and the graph isomorphism problem, see [,10,15] E-mail address: huabai@uscedu H Bai 004-3795/03/$ - see front matter 003 Published by Elsevier Science Inc doi:101016/s004-379500077-9

5 H Bai / Linear Algebra and its Applications 369 003 51 61 When G is connected, the kernel of LG is spanned by the all-1 vector 1, 1,,1 t in R n, where superscript t denotes the transpose Considering LG as a linear map from Z n to itself, its cokernel has the form Z n /Im LG Z KG, where KG is defined to be the critical group also called the Picard group [1], Jacobian group [3] or sandpile group [6] It follows from Kirchhoff s matrix-tree theorem that the order KG is known to be KG, the number of spanning trees in G Kirchhoff s Matrix-Tree Theorem see, eg [, Chapter 6] i KG 1 i+j det LG here LG is a reduced Laplacian matrix obtained from LG by deleting row i and column j ii For any graph G with n vertices, index the eigenvalues of the Laplacian LG in weakly decreasing order: λ 1 λ λ n 1 λ n 0 then KG λ 1λ λ n 1 n Our main tools will be the use of Smith Normal Form for integer matrices Given a square integer matrix A, its Smith normal form is the unique equivalent diagonal matrix SA diag[s 1,s,,s n ] whose entries s i are nonnegative and s i divides s i+1 Thes i are known as the invariant factors of A [16] Two integral matrices A and B are equivalent denoted as A B if there exist integer matrices P and Q of determinant ±1 that satisfy PAQ B The structure of the critical group is closely related with the Laplacian matrix: if the Smith normal form of LG is diag[s 1,s,,s n ],thenkg is the torsion subgroup of Z s1 Z s Z sn HereZ n denote the cyclic group Z/nZ for any nonnegative integer n There are very few families of graphs for which the critical group structure has been completely determined, such as complete graphs and complete bipartite graphs [13], cycles [14], wheels [4], generic threshold graphs [5], etc Reiner [17] proposed two conjectures on the structure of the critical group of n-cube KQ n His first conjecture is the following theorem, which we prove in Section Theorem 11 The critical group of Q n has exactly n 1 1 invariant factors, for n 1

H Bai / Linear Algebra and its Applications 369 003 51 61 53 We show the structure of the Sylow p-group of the critical group of the n-cube when p is an odd prime Theorem 1 For any odd prime number p, the Sylowp-group of thecritical group of the n-cube Syl p KQ n has the following expression: n Syl p KQ n Syl p Z n k k k1 Recently Jacobson, Niedermaier and Reiner generalize this result to the Cartesian product of complete graphs K n1 K nk Jacobson Niedermaier Reiner Theorem [1, Theorem ] For every prime p that divides none of n 1,,n k, the Sylow p-subgroup or p-primary component of the critical group KK n1 K nk has the following description: Syl p KK n1 K nk Syl p Z NS i S ni 1, where N S : i S n i /S [k] One knows from simple eigenvalue calculations see [1, Section ] that 1 i S KK n1 K nk ki1 N n i 1 S n i /S [k] Note that this general result on p-primary structures for p that divides none of n 1,,n k is as simple as one could hope Our next result is related to the -primary structure of KQ n Let a n denotes the number of occurrences of Z in the elementary divisor form of the critical group of the n-cube KQ n Reiner s second conjecture is confirmed in Section 4 Theorem 13 The generating function of a n is a n+3 x n 1 1 x1 x n0 This leads to a simple expression for a n a 0 a 1 0: a n n n / when n For the background of the elementary divisor form of a finitely generated abelian group see [8, p 163] The full structure of the Sylow- subgroup of the critical group of the n-cube is still unknown

54 H Bai / Linear Algebra and its Applications 369 003 51 61 Some lemmas and the proof of Theorem 11 Let L n LQ n Clearly L 0 0 and since Q n Q n 1 Q 1 for n 1, one can order the vertices of Q n such that Ln 1 + 1 1 L n 1 L n 1 + 1 The size of L n is n, which is the number of vertices of Q n Throughout this paper let L n,k : L n + k, n and k be nonnegative integers, where k denotes the scalar matrix ki with I an identity matrix of same size as L n In general, if A is a matrix and c a constant, A + c denotes A + ci IfAand B are two square matrices, we also make following convention: A B : A 0 0 B and A k : k times {}}{ A A A Let M n R denote the set of n n matrices with entries in the ring R Ifthesize of the matrices does not matter or is clear, sometimes we simply write M n R as MR Lemma 1 1 L n,k+i MZ m! for any m N In particular,l n L n + MZ Proof Let I n 1 I T n : n 1, L I n 1 I n 1,k : L n 1,k L n 1,k n 1 Then L n 1,k commutes with T n,and 1 T n 1 T n Hence 1 j 1 j L n,k L n 1,k + T n 1 j L n 1,k + 1 T j 1 j 1 l n l L n 1,k Note that x + i 1 j L n 1,k + 1 T n m cm, jx j j1 l0 1 + 1 L n 1,k j 1 j L n 1,k

H Bai / Linear Algebra and its Applications 369 003 51 61 55 where cm, j is the signless Stirling number of the first kind, L n,k+i m j1 1 j m cm, j L n,k L n 1,k+i + 1 T n L n 1,k+i + L n 1,k+i + mt n i1 L n 1,k+i L n 1,k+i Now by induction on n + m the lemma is trivial when n 0sinceL 0 0, 1 L n 1,k+i m! is in MZ, andsois 1 1 mt n L n 1,k+i T n 1 L n 1,k+1+i m! m 1! i1 So the lemma is true for all n, m Lemma L n+1,k I n L n,k L n,k+1 Consequently, we have a group isomorphism Z n+1/ Im L n+1,k Z n/ Im L n,k L n,k+1 Proof 1 + Ln,k 1 L n+1,k 1 1+ L n,k 0 1 L n,k L n,k + 1 + L n,k 1 0 0 L n,k L n,k+1 Corollary 3 The Smith normal form of L n has exactly n 1 occurrences of 1, for all n 1 Proof Let k 0 and replace n by n 1 in the above lemma Note that there is no occurrence of 1 in the Smith normal form of L n 1 L n 1 +, which is in M n 1Z Hence it concludes

56 H Bai / Linear Algebra and its Applications 369 003 51 61 This statement is equivalent to cf Fiol [9] the following theorem, which is Reiner s first conjecture on the number of invariant factors of KQ n : Theorem 11 The critical group of Q n has exactly n 1 1 invariant factors 3 The Sylow p-group of KQ n for p/ In this section we will determine the structure of the Sylow p-group of the critical group KQ n for all odd primes p If A is an n n integer matrix, let KA be the torsion subgroup of the finitely generated abelian group Z n /Im A IfG is an abelian group, let Syl p G denote the Sylow p-group of G Proposition 31 Syl p KL n+1,k Syl p KL n,k Syl p KL n,k+1 Proof We have two group homomorphisms ψ and φ: KL n,k L n,k+1 ψ φ KL n,k KL n,k+1 ; ψ w + Im L n,k L n,k+1 : w + Im Ln,k,w+ Im L n,k+1, φ u + Im L n,k,v+ Im L n,k+1 : Ln,k+1 u L n,k v + Im L n,k L n,k+1 Let G denote the multiplication by homomorphism of G for any abelian group G It is straightforward to check that ψ φ KLn,k KL n,k+1 and φ ψ KLn,k L n,k+1 Now consider the restriction of φ and ψ to Sylow p-subgroups for p an odd prime Because G is an isomorphism for any finite abelian p-group G, both ψ and φ must be isomorphisms, so Syl p KL n+1,k Syl p KL n,k Syl p KL n,k+1 Theorem 1 For any odd prime number p, the Sylowp-group of thecritical group of the n-cube Syl p KQ n has the following expression: n Syl p KQ n Syl p Z n k k k1 Proof Let us compute the Sylow p-group of KQ n recursively: Syl p KQ n Syl p KL n 1,0 Syl p KL n 1,1

H Bai / Linear Algebra and its Applications 369 003 51 61 57 Syl p KL n,0 Syl p KL n,1 Sylp KL n, j Sylp KL n j,k j k k0 n Sylp KL 0,k n k k0 let j n In the final step, Syl p KL 0,k by definition is Syl p Z 0 /ImL 0 + k, which is just Syl p Z k, thus we obtain n Syl p KQ n Syl p Z n k k k1 4 Proof of Theorem 13 In this section we get a partial decomposition of Syl KQ n, which leads to the proof of Reiner s second conjecture on the occurrences of Z in Syl KQ n Lemma 41 If m is an even positive integer, then m Ln,k L L n+,k+i L n,k+i L n,k+1 m n,k+i 0 L n,k+m L n,k+m+1 i1 i in M n+z N, where N is any positive integer Proof Recall that by Lemma 1 we have: L n,k+i i1 L n 1,k+i + mt n i1 Ln 1,k + m L n 1,k+i L n 1,k+i L n 1,k + m We will try to do some elementary row and column operations [11] to get desired decomposition of the Smith normal form These operations consist of: adding an integer multiple of one row or column to another; negating a row or column; interchanging two rows or columns Here we will use the theory of the Smith normal form for matrices with entries in a principal idea domain cf [7], in this case it is Z N L n+,k+i i1 Ln+1,k Γ11 Γ L n+1,k+i 1 0 L n+1,k+m 0 Γ,

58 H Bai / Linear Algebra and its Applications 369 003 51 61 where Γ 11 i1 Γ 1 Γ Ln,k + m L n,k+i i, L n,k + m Ln,k+1 + m 1 1 L n,k+i, 1 L n,k+1 + m 1 m Ln,k+1 + m L n,k+i i L n,k+1 + m By the same sort of operations, it is easy to see that where Γ11 Γ 1 0 Γ Λ i L n,k+i 4 Λ, L n,k L n,k+1 L n,k+1 L n,k+1 mm 1 0 L n,k+1 L n,k+m 0 L n,k+m 0 0 L n,k+1 L n,k+m L n,k+m 0 0 0 L n,k+m L n,k+m+1 Since m is even as assumption, there exists an integer r such that m 1r 1mod N Similarly, there exists an integer s making 1 mrs 1mod N,or, equivalently, 1 + mrs smod N Let P,Qbe block matrices in SL n+z N as 1 0 0 0 rsl P n,k+m 1 + rsm rsm 0 r sl n,k+m rs rs 0 0 L n,k+m+1 L n,k+m+1 1 mrm 1 1 0 0 0 rsl Q n,k 1 0 0 rsl n,k 1 m 1 0 r sl n,k L n,k+1 0 L n,k+1 r, Then we can check that in MZ N L n,k L n,k+1 0 0 m P ΛQ 0 L n,k+1 L n,k+m 0 0 0 0 L n,k+1 L n,k+m 0 0 0 0 L n,k+m L n,k+m+1 Thinking of the coefficient i L n,k+i 4, we get the desired decomposition of the Smith normal form in the lemma

H Bai / Linear Algebra and its Applications 369 003 51 61 59 In the special case that m, we already know that 1 L n,kl n,k+1 is still an integer matrix, so we may continue elementary row/column operations: Ln,k L n,k+1 0 L n,k+ L n,k+3 0 1 L n,kl n,k+1 L n,k+ L n,k+3 L n,k+ L n,k+3 I n 1 3 L n,k+i So we have only in MZ N : L n+,k L n+,k+1 I n L n,k+1 L n,k+ 1 3 L n,k+i Knowing that KL n+1,k KL n,k L n,k+1 Lemma, we can rewrite this as Corollary 4 Syl KL n,k Z n 3 Syl KL n,k+1 Syl K for any n 3 1 3 L n 3,k+i, Let an, k be the number of the occurrences of Z in Syl KL n,k In particular Reiner s second conjecture concerns of a n : an, 0 for Syl KQ n Note that Z does not show up in Syl K 1 3 L n,k+i for any n and k, since the matrix 3 L n,k+i is in M4Z by Lemma 1 Now we can confirm Reiner s second conjecture: Theorem 13 The generating function of a n is a n+3 x n 1 1 x1 x n0 Proof By Corollary 4, an, k has recurrence an, k n 3 + an,k+ 1, n 3 We evaluate directly the initial conditions: a0,k a1,k a,k 0, k 0 These initial conditions imply that an, k a n

60 H Bai / Linear Algebra and its Applications 369 003 51 61 Let fx n0 a n+3 x n,then fx n x n + a n+1 x n n0 n0 n0 x n + a n+1 x n n x n + x a n+3 x n n0 n0 1 1 x + x fx Hence we conclude that fx 1/[1 x1 x ] Reiner shares with the author the data computed by the Smith normal form program at web site http://linboxpccisudeledu:8080/gap/smithformhtml n Syl KQ n Z 4 3 Z Z 8 4 Z Z4 8 Z 3 5 Z 6 Z4 8 Z 16 Z 4 64 6 Z 1 Z4 4 Z 8 Z 4 3 Z10 64 7 Z 8 Z 4 Z 8 16 Z6 3 Z14 64 Z6 18 8 Z 56 Z 4 Z16 16 Z1 3 Z8 64 Z1 18 Z 104 9 Z 10 Z 10 4 Z16 16 Z6 3 Z48 64 Z6 18 Z 51 Z 8 048 10 Z 40 Z 36 4 Z6 8 Z16 3 Z148 64 Z 56 Z 6 104 Z18 048 11 Z 496 Z 66 4 Z3 8 Z100 16 Z164 64 Z 18 Z 100 51 Z64 048 Acknowledgments The author is indebted to Victor Reiner for drawing my attention to the structure of the critical group of the n-cube His suggestions and discussions also greatly helped in improving the final version of this paper The author also thanks the referee for valuable comments

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