Critical groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields: an extended abstract
aa r X i v : . [ m a t h . C O ] D ec CRITICAL GROUPS OF GENERALIZED DE BRUIJN AND KAUTZGRAPHS AND CIRCULANT MATRICES OVER FINITE FIELDS
SWEE HONG CHAN, HENK D. L. HOLLMANN, DMITRII V. PASECHNIK
School of Physical and Mathematical Sciences, Nanyang TechnologicalUniversity, 21 Nanyang Link, Singapore 637371 A BSTRACT . We determine the critical groups of the generalized de Bruijngraphs
DB( n, d ) and generalized Kautz graphs Kautz( n, d ) , thus extendingand completing earlier results for the classical de Bruijn and Kautz graphs.Moreover, for a prime p the critical groups of DB( n, p ) are shown to be inclose correspondence with groups of n × n circulant matrices over F p , whichexplains numerical data in [11], and suggests the possibility to construct nor-mal bases in F p n from spanning trees in DB( n, p ) .
1. I
NTRODUCTION
The critical group of a directed graph G is an abelian group obtained from theLaplacian matrix ∆ of G ; it determines and is determined by the Smith NormalForm (SNF) of ∆ . (For precise definitions of these and other terms, we refer tothe next section.) The sandpile group S ( G, v ) of G at a vertex v is an abeliangroup obtained from the reduced Laplacian ∆ v of G ; its order is equal to the complexity κ ( G ) of G , the number of directed trees rooted at v , a fact that isrelated to the Matrix Tree Theorem, see for example [8] and its references. If G is Eulerian, then S ( G, v ) does not depend on v , and is then simply writtenas S ( G ) ; in that case, it is equal to the critical group of G . The critical grouphas been studied in other contexts under several other names, such as group ofcomponents, Picard or Jacobian group, and Smith group. For more details andbackground, see, e.g., [6].Critical groups have been determined for a large number of graph fami-lies. For some examples, see the references in [1]. Here, we determine thecritical group of the generalized de Bruijn graphs DB( n, d ) and generalizedKautz graphs Kautz( n, d ) , thus extending and completing the results from [8]for the binary de Bruijn graphs DB(2 ℓ , and Kautz graphs (with p prime) Kautz(( p − p ℓ − , p ) , and [3] for the classical de Bruijn graphs DB( d ℓ , d ) and Kautz graphs Kautz(( d − d ℓ − , d ) . Unlike the classical case, the gener-alized versions are not necessarily iterated line graphs, so to obtain their criticalgroups, different techniques have to be applied. E-mail address : { sweehong, henk.hollmann, dima } @ntu.edu.sg . Our original motivation for studying these groups stems from their relationsto some algebraic objects, such as the groups C ( n, p ) of invertible n × n -circulant matrices over F p (mysterious numerical coincidences were noted inthe OIES entry A027362 [11] by the third author, computed with the help of[15, 12]), and normal bases (cf. e.g. [9]) of the finite fields F p n . The latter werenoted to be closely related to circulant matrices and to necklaces by Reutenauer[13, Sect. 7.6.2], see also [5], and the related numeric data collected in [2]. Inparticular, we show that C ( n, p ) / ( Z p − × Z n ) is isomorphic to the critical groupof DB( n, p ) . Although we were not able to construct an explicit bijection be-tween the former and the latter, we could speculate that potentially one might beable to design a new deterministic way to construct normal bases of F p n .2. P RELIMINARIES
Let M be an m × n integer matrix of rank r . For a ring F , we write R F ( M ) = M ⊤ F n , the F -module generated by the rows of M . The Smith group [14] of M is defined as Γ( M ) = Z n /R Z ( M ) . The submodule Γ( M ) = Z n /R Q ( M ) ∩ Z n of Γ( M ) is a finite abelian group called the finite part of Γ( M ) . Indeed, if M hasrank r , then Γ( M ) = Z n − r ⊕ Γ( M ) with Γ( M ) = ⊕ ri =1 Z d i , where d , . . . , d r are the nonzero invariant factors of M , so that d i | d i +1 for i = 1 , . . . , r − .For invariant factors and the Smith Normal Form, we refer to [10]. See [14] forfurther details and proofs.Let G = ( V, E ) be a directed graph on n = | V | vertices. The indegree d − ( v ) and outdegree d + ( v ) is the number of edges ending or starting in v ∈ V ,respectively. The adjacency matrix of G is the n × n matrix A = ( A v,w ) , withrows and columns indexed by V , where A v,w is the number of edges from v to w . The Laplacian of G is the matrix ∆ = D − A , where D is diagonalwith D v,v = d − v . The critical group K ( G ) of G is the finite part of the Smithgroup of the Laplacian ∆ of G . The sandpile group S ( G, v ) of G at a v ∈ V isthe finite part of the Smith group of the ( n − × ( n − reduced Laplacian ∆ v , obtained from ∆ by deleting the row and the column of ∆ indexed by v .Note that by the Matrix Tree Theorem for directed graphs, the order of S ( G, v ) equals the number of directed spanning trees rooted at v . G is called Eulerian if d + ( v ) = d − ( v ) for every v ∈ V . In that case, S ( G, v ) does not depend on v andis equal to the critical group S ( G ) of G . For more details on sandpile groupsand critical groups of directed graphs, we refer for example to [6] or [16].2.1. Generalized de Bruijn and Kautz graphs.
Generalized de Bruijn graphsand generalized Kautz graphs [4] are known to have a relatively small diameterand attractive connectivity properties, and have been studied intensively due totheir applications in interconnection networks. The generalized Kautz graphswere first investigated in [7], and are also known as
Imase-Itoh digraphs . Bothclasses of graphs are Eulerian.We will determine the critical group, or, equivalently, the sandpile group, ofa generalized de Bruijn or Kautz graph on n vertices by embedding this group RITICAL GROUPS OF DE BRUIJN AND KAUTZ GRAPHS AND CIRCULANTS 3 as a subgroup of index n in a group that we will refer to as the sand dune groupof the corresponding digraph. Let us now turn to the details.The generalized de Bruijn graph DB( n, d ) has vertex set Z n , the set of in-tegers modulo n , and (directed) edges v → dv + i for i = 0 , . . . , d − andall v ∈ Z n . The generalized Kautz graph Kautz( n, d ) has vertex set Z n anddirected edges v → − d ( v + 1) + i for i = 0 , . . . , d − and all v ∈ Z n . Note thatboth DB( n, d ) and Kautz( n, d ) are Eulerian. In what follows, we will focus onthe generalized de Bruijn graph; the generalized Kautz graph can be handled ina similar way, essentially by replacing d by − d in certain places.Let Z n = { a ( x ) ∈ Z [ x ] mod x n − | a (1) = 0 } . With each vertex v ∈ Z n ,we associate the polynomial f v ( x ) = dx v − x dv P d − i =0 x i ∈ Z n . Since f v ( x ) isthe associated polynomial of the v th row of the Laplacian ∆ ( n,d ) of the general-ized de Bruijn graph DB( n, d ) , the Smith group Γ(∆ ( n,d ) ) of the Laplacian of DB( n, d ) is the quotient of Z [ x ] mod x n − by the Z n -span h f v ( x ) | v ∈ Z n i Z n of the polynomials f v ( x ) . Now note that Z [ x ] mod x n − ∼ = Z ⊕ Z n , so since P v ∈ Z n f v ( x ) = 0 , we have that Γ(∆ ( n,d ) ) = ( Z [ x ] mod x n − / h f v ( x ) | v ∈ Z n i Z n (1) ∼ = Z ⊕ Z n / h f v ( x ) | v ∈ Z ′ n i Z n where Z ′ n = Z n \ { } . It is easily checked that the polynomials f v ( x ) with v ∈ Z ′ n are independent over Q , hence they constitute a basis for Z n over Q . Asa consequence, each element in the quotient group(2) S ( n, d ) = S DB ( n, d ) = Z n / h f v ( x ) | v ∈ Z ′ n i Z n has finite order, and so S ( n, d ) is the critical group, or, equivalently, the sandpilegroup, of the generalized de Bruijn graph DB( n, d ) . We define the sand dunegroup Σ( n, d ) = Σ DB ( n, d ) of DB( n, d ) as Σ( n, d ) = Z n / h g v ( x ) | v ∈ Z ′ n i Z n ,where g v ( x ) = ( x − f v ( x ) = dx v ( x − − x dv ( x d − . Now let e v = x v − ; we have that e = 0 , and Z n = h e v | v ∈ Z ′ n i Z , the Z -span ofthe polynomials e v . Furthermore, let ǫ v = de v − e dv . The span in Q n = { a ( x ) ∈ Q [ x ] mod x n − | a (1) = 0 } of the polynomials g v ( x ) with v ∈ Z n is the set of polynomials of the form dc ( x ) − c ( x d ) with c (1) = 0 ; since ǫ v = g ( x ) + · · · g v − ( x ) for all v ∈ Z n , we conclude that(3) Σ( n, d ) = Z n / E n,d , where Z n = h e v | v ∈ Z ′ n i Z and E n,d = h ǫ v | v ∈ Z ′ n i Z is the Z -submodule of Z n generated by the polynomials ǫ v = de v − e dv . The next result is crucial: itidentifies the elements of the sand dune group Σ( n, d ) that are actually containedin the sandpile group S ( n, d ) . (Due to lack of space, we omit the not too difficultproofs in the remainder of this section.) Theorem 2.1. If a ∈ Σ( n, d ) with a = P v a v e v , then a ∈ S ( n, d ) if and onlyif P v va v ≡ n . Corollary 2.2.
We have Σ( n, d ) /S ( n, d ) = Z n and so | Σ( n, d ) | = n | S ( n, d ) | . CRITICAL GROUPS OF DE BRUIJN AND KAUTZ GRAPHS AND CIRCULANTS
The above descriptions of the sandpile group S ( n, d ) and sand dune group Σ( n, d ) , and the embedding of S ( n, d ) as a subgroup of Σ( n, d ) are very suitablefor the determination of these groups. In the process, repeatedly information isrequired about the order of various group elements. The following two resultsprovide that information. Lemma 2.3.
Let a = P v a v ǫ v ∈ Σ( n, d ) . Then the order of a in Σ( n, d ) is thesmallest positive integer m for which ma v ∈ Z for each v . We say that v ∈ Z n has d -type ( f, e ) in Z n if v, dv, . . . , d e + f − v are alldistinct, with d e + f v = d f v . Now, by expressing e v in terms of the ǫ v , we candetermine the order of e v . The result is as follows. Lemma 2.4.
Supposing v has d -type ( f, e ) , then e v = P f − i =0 d − i − ǫ d i v + P e − j =0 d j − f ( d e − − ǫ d f + j v in Z n , and hence e v has order d f ( d e − in Σ( n, d ) . Invertible circulant matrices.
Let Q n be the n × n permutation matrixover a field F corresponding to the cyclic permutation (1 , , . . . , n ) . An n × n circulant matrix over F is a matrix that can be written as a Q n + a Q n + . . . + a n Q nn with a i ∈ F for ≤ i ≤ n . All the invertible circulant matrices form acommutative group (w.r.t. matrix multiplication), namely, the centralizer of Q n in GL n ( F ) . In the case F = F p we consider here we denote this commutativegroup by C ( n, p ) . Note that C ( n, p ) contains a subgroup isomorphic to Z p − ⊕ Z n , namely the direct product of the group of scalar matrices F ∗ p I := { λI | λ ∈ F ∗ p } and the cyclic subgroup generated by Q n . Each circulant matrix hasall-ones vector := (1 , . . . , ⊤ as an eigenvector. Thus C ′ ( n, p ) := { g ∈ C ( n, p ) | g = } is a subgroup of C ( n, p ) , and we have the following formula.(4) C ( n, p ) = C ′ ( n, p ) × F ∗ p I.
3. M
AIN RESULTS
Let n, d > be fixed integers. The description of the sandpile group S ( n, d ) and the sand-dune group Σ( n, d ) of the generalized the Bruin graph DB( n, d ) involves a sequence of numbers defined as follows. Put n = n , and for i =1 , , . . . , define g i = gcd( n i , d ) and n i +1 = n i /g i . We have n > · · · > n k = n k +1 , where k is the smallest integer for which g k = 1 . We will refer to thesequence n > · · · > n k = n k +1 as the d -sequence of n . In what follows, wewill write m = n k and g = g · · · g k − . Note that n = gm with gcd( m, d ) = 1 .Since gcd( m, d ) = 1 , the map x → dx partitions Z m into orbits of the form O ( v ) = ( v, dv, . . . , d o ( v ) − v ) . We will refer to o ( v ) = | O ( v ) | as the order of v .For every prime p | m , we define π p ( m ) to be the largest power of p dividing m . Let V be a complete set of representatives of the orbits O ( v ) different from { } , where we ensure that for every divisor p of m , all integers of the form m/p j are contained in V . RITICAL GROUPS OF DE BRUIJN AND KAUTZ GRAPHS AND CIRCULANTS 5
Theorem 3.1.
With the above definitions and notation, we have that (5) Σ( n, d ) = (cid:20) k − M i =0 Z n i − n i +1 + n i +2 d i +1 (cid:21) ⊕ (cid:20)M v ∈ V Z d o ( v ) − (cid:21) , and (6) S ( n, d ) = (cid:20) k − M i =0 Z d i +1 /g i ⊕ Z n i − n i +1 + n i +2 − d i +1 (cid:21) ⊕ "M v ∈ V Z ( d o ( v ) − /c ( v ) , where c ( v ) = 1 except in the following cases. For any p | m , c ( m/π p ( m )) = (cid:26) π p ( m ) , if p = 2 or d ≡ or m ; π ( m ) / , if p = 2 and d ≡ and | m, and if | m and d ≡ , then c ( m/
2) = 2 . For the generalized Kautz graph, a similar result holds. For v ∈ Z m , we let O ′ ( v ) denote the orbit of v under the map x → − dv , and we define o ′ ( v ) = | O ′ ( v ) | . Now take V ′ to be a complete set of representatives of the orbits on Z ′ m . Finally, define c ′ ( v ) similar to c ( v ) , except that now d is replaced by − d (so the special case now involves d ≡ ). Then we have the following. Theorem 3.2.
The sandpile group S Kautz ( n, d ) of the generalized Kautz graph Kautz( n, d ) is obtained from S ( n, d ) by replacing V by V ′ , o ( v ) by o ′ ( v ) , and c ( v ) by c ′ ( v ) in (6). The above results can be proved in a number of steps. In what follows, weoutline the method for the generalized de Bruijn graphs; for the generalizedKautz graphs, a similar approach can be used. Furthermore, we note that manyof the steps below repeatedly use Theorem 2.1 and Lemma 2.4. First, we inves-tigate the “multiplication-by- d ” map d : x → dx on the sandpile and sand-dunegroup. Let Σ ( n, d ) and S ( n, d ) denote the kernel of the map d k on Σ( n, d ) and S ( n, d ) , respectively. It is not difficult to see that Σ( n, d ) ∼ = Σ ( n, d ) ⊕ Σ( m, d ) and S ( n, d ) ∼ = S ( n, d ) ⊕ S ( m, d ) . Then, we use the map d to determine Σ ( n, d ) and S ( n, d ) . It is easy to see that for any n , we have d Σ( n, d ) ∼ =Σ( n/ ( n, d ) , d ) and dS ( n, d ) ∼ = S ( n/ ( n, d ) , d ) . With much more effort, it canbe show that the kernel of the map d on Σ( n, d ) and S ( n, d ) is isomorphic to Z n − n/ ( n,d ) d and Z d/ ( n,d ) ⊕ Z n − − n/ ( n,d ) d , respectively. Then we use inductionover the length k + 1 of the d -sequence of n to show that Σ ( n, d ) and S ( n, d ) have the form of the left part of the right hand side in (5) and (6), respectively.This part of the proof, although much more complicated, resembles the methodused by [8] and [3].Now it remains to handle the parts Σ( m, d ) and S ( m, d ) with gcd( m, d ) = 1 .For the “helper” group Σ( m, d ) that embeds S ( m, d ) , this is trivial: it is easilyseen that Σ( m, d ) = ⊕ v ∈ V h e v i , and the order of e v is equal to the size o ( v ) of its orbit O ( v ) under the map d , so (5) follows immediately. The e v are notcontained in S ( m, d ) , but we can try to modify them slightly to obtain a similardecomposition for S ( m, d ) . The idea is to replace e v by a modified version CRITICAL GROUPS OF DE BRUIJN AND KAUTZ GRAPHS AND CIRCULANTS ˜ e v = e v − P p | m λ p ( v ) e π p ( v ) m/π p ( m ) , where the numbers λ p ( v ) are chosensuch that ˜ e v ∈ S ( m, d ) , or by a suitable multiple of e v , in some exceptionalcases (these are cases where c ( v ) > ). It turns out that this is indeed possible,and in this way the proof of Theorem 3.1 can be completed.Finally, with the notation from Subsect. 2.2, we have the following isomor-phisms, connecting critical groups and circulant matrices. Theorem 3.3.
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