On Moore Bipartite Digraphs

Transcription

1 On Moore Bipartite Digraphs M. A. Fiol, 1 * J. Gimbert, 2 J. Gómez, 1 and Y. Wu 3 1 DEPARTMENT DE MATEMÀTICA APLICADA IV TELEMÀTICA UNIVERSITAT POLITÈCNICA DE CATALUNYA JORDI GIRONA 1-3, MÒDUL C3 CAMPUS NORD BARCELONA, SPAIN fiol@mat.upc.es 2 DEPARTMENT DE MATEMÀTICA UNIVERSITAT DE LIEIDA JAUME II 69, LLEIDA, SPAIN 3 DEPARTMENT OF MATHEMATICS SHANGHAI JIAO TONG UNIVERSITY 1954 HUASHAN ROAD SHANGHAI , CHINA Received March 1, 2000; Revised December 3, 2002 DOI /jgt Abstract: In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore-like bound in terms of its diameter k and the maximum out-degrees (d 1 ; d 2 ) of its partite sets of vertices. It has been proved that, when d 1 d 2 > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 k 4. This paper deals with *Correspondence to: Miguel Angel Fiol, Department de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Jordi Girona 1-3, Edifici C3, Campus Nord, Barcelona, Spain. fiol@mat.upc.es ß 2003 Wiley Periodicals, Inc.

2 172 JOURNAL OF GRAPH THEORY the problem of their enumeration. In this context, using the theory of circulant matrices and the so-called De Bruijn near-factorizations of cyclic groups, we present some new constructions of Moore bipartite digraphs of diameter three and composite out-degrees. By applying the iterated line digraph technique, such constructions also provide new families of dense bipartite digraphs with arbitrary diameter. Moreover, we show that the line digraph structure is inherent in any Moore bipartite digraph G of diameter k ¼ 4, which means that G ¼ LG 0, where G 0 is a Moore bipartite digraph of diameter k ¼ 3. ß 2003 Wiley Periodicals, Inc. J Graph Theory 43: , 2003 Keywords: bipartite digraph; Moore bound; circulant matrix; De Bruijn near-factorization; line digraph 1. INTRODUCTION For its implications in the design of interconnection networks, it is interesting to find families of dense digraphs, that is to say, strongly connected digraphs with a relatively large number of vertices in comparison with the largest order allowed by their maximum out-degree and diameter. In the case of bipartite digraphs with maximum out-degrees ðd 1 ; d 2 Þ, not necessarily equal in both partite sets of vertices ðv 1 ; V 2 Þ, the order N is bounded above by the Moore-like bound N Mðd 1 ; d 2 ; kþ ¼ 2ð1 þ d 1d 2 þþðd 1 d 2 Þ m Þ if k ¼ 2m þ 1; ðd 1 þ d 2 Þð1 þ d 1 d 2 þþðd 1 d 2 Þ m 1 Þ if k ¼ 2m; where k stands for the diameter. To derive this bound, we simply consider that, from any vertex v, all vertices in the same (respectively, different) partite set of v must be reached in at most k 1 steps, if k is odd (respectively, even); see [9]. A bipartite digraph G ¼ðV; EÞ, V ¼ V 1 [ V 2, with maximum out-degrees ðd 1 ; d 2 Þ, diameter k, and (optimal) order N ¼ Mðd 1 ; d 2 ; kþ is called a Moore bipartite digraph. In fact, such a digraph must be semi-out-regular with out-degrees ðd 1 ; d 2 Þ (or ðd 1 ; d 2 Þ-outregular for short); that is, d þ ðvþ ¼d for any v 2 V (throughout the paper, 2f1; 2g). The problem of the Moore bipartite digraphs existence has already been solved in the regular case (d 1 ¼ d 2 ) by Fiol and Yebra [9], and their results can be easily extended to the general case, as was noted by Gómez, Padró, and Perennes [12] in the context of generalized cycles. Thus, it is known that the bound Mðd 1 ; d 2 ; kþ is only attained when 2 k 4, if d 1 d 2 > 1. The non-existence of a Moore bipartite digraph G ¼ðV 1 [ V 2 ; EÞ with diameter k > 4, out-degrees ðd 1 ; d 2 Þ, d 1 d 2 > 1, and order N ¼ n 1 þ n 2, n : ¼jV j¼ 1 þ d 1d 2 þþðd 1 d 2 Þ m if k ¼ 2m þ 1; d ð1 þ d 1 d 2 þþðd 1 d 2 Þ m 1 ð2þ Þ if k ¼ 2m; ð1þ

3 (where 1 ¼ 2 and 2 ¼ 1Þ, has been deduced by applying spectral techniques to the equation that its adjacency matrix A ¼ bip ða 1 ; A 2 Þ :¼ 0 A 1 A 2 0 ON MOORE BIPARTITE DIGRAPHS 173 ð3þ must satisfy. Depending on the parity of k, such an equation is I þ A þþa 2mþ1 J ¼ n d 2 J n d 1 J n J n if k ¼ 2m þ 1 ð4þ (in this case n :¼ n 1 ¼ n 2 ; see (2)), and A þ A 2 þþa 2m ¼ d 1J n1 J n1 ;n 2 J n2 ;n 1 d 2 J n2 if k ¼ 2m; ð5þ where J denotes the all-one matrix with the appropriate dimensions. In matricial terms, these equations formulate two essential properties of G. Namely, the uniqueness of a walk of length k 1 between each pair of vertices in the same or different partite set of VðGÞ, depending upon the parity of k; and the semi-outregularity of the digraph. Using the expression (3) for A, such equations are equivalent to A J ¼ d J (in which case we say that A is d -outregular) and I n þ A 1 A 2 þþða 1 A 2 Þ m ¼ J n I n þ A 2 A 1 þþða 2 A 1 Þ m if k ¼ 2m þ 1; ð6þ ¼ J n ) A 1 ði n2 þ A 2 A 1 þþða 2 A 1 Þ m 1 Þ¼J n1 ;n 2 if k ¼ 2m; ð7þ A 2 ði n1 þ A 1 A 2 þþða 1 A 2 Þ m 1 Þ¼J n2 ;n 1 respectively; see [9,12]. Nevertheless, the enumeration of Moore bipartite digraphs remains unsettled, apart from the trivial cases k ¼ 2, and d 1 ¼ d 2 ¼ 1(k odd), which have unique solutions: the complete bipartite digraph K d1 ;d 2 and the directed cycle of length k þ 1, respectively. For diameter k ¼ 3, only one construction was known. Namely, the family of Moore bipartite digraphs BDðd 1 ; d 2 ; nþ, where n :¼ n ¼ c1 þ d 1 d 2, introduced in [12] as a natural extension of the regular Moore bipartite digraphs BDðd; nþ presented in [9]; see also [2]. A digraph BDðd 1 ; d 2 ; nþ has set of vertices V ¼ V 1 [ V 2, where V ¼fgZ n, and the vertices ð1; iþ and ð2; iþ are, respectively adjacent to the vertices of the sets N þ ð1; iþ ¼fð2; d 1 i þ tþ; t ¼ 0; 1;...; d 1 1g; N þ ð2; iþ ¼fð1; d 2 ði þ 1ÞþtÞ; t ¼ 0; 1;...; d 2 1g: ð8þ ð9þ

4 174 JOURNAL OF GRAPH THEORY (From now on, and unless otherwise stated, arithmetic must be understood in the considered ring, e.g., Z n.) Then, De Bruijn type adjacencies (8), from V 1 to V 2, are represented by a binary submatrix A 1 in which each row (excluding the first one) is the previous row moved to the right d 1 places, with wraparound; that is, A 1 is a d 1 -circulant matrix. Notice that A 1 is uniquely determined by its shifting parameter d 1 and its first row ða 0 ; a 1 ;...; a n 1 Þ¼ð1;... d 1 ; 1; 0;...; 0Þ. This can be represented by the polynomial A1 ðxþ :¼ Xn 1 a i x i ¼ 1 þ x þþx d1 1 ; i¼0 which is called the Hall polynomial of A 1. Analogously, Kautz type adjacencies (9), from V 2 to V 1, are described by a ð d 2 Þ-circulant binary submatrix A 2 with Hall polynomial A2 ðxþ ¼x n d 2 þ x n d 2þ1 þþx n 1, since N þ ð2; 0Þ ¼ fð1; d 2 þ tþ; t ¼ 0; 1;...; d 2 1g. This fact suggests seeking other solutions to (6), in the case m ¼ 1(k ¼ 3), using the theory of circulant matrices. In this context, the key idea is to use the known isomorphism between the subring of circulant matrices of order n with rational coefficients and the quotient ring Q½xŠ=ðx n 1Þ, which assigns to each matrix its Hall polynomial; see e.g. [5]. Thus, in Section 2, we reformulate such a search in polynomial and additive terms, which allows us to construct new Moore bipartite digraphs of diameter three and composite out-degrees by using the so-called De Bruijn nearfactorizations of cyclic groups. From such optimal digraphs, and using the iterated line digraph technique introduced in [7,8], we can obtain new families of dense bipartite digraphs with arbitrary diameter. We point out that, from the results in [7,8], if G is a Moore bipartite digraph with diameter k ¼ 3 and out-degrees ðd 1 ; d 2 Þ, d 1 d 2 > 1, then its ð2r 1Þ-th iterated line digraph L 2r 1 G, r 1, is a ðd 1 ; d 2 Þ-outregular bipartite digraph of diameter k ¼ 2ðr þ 1Þ and order N ¼ðd 1 þ d 2 Þððd 1 d 2 Þ r þðd 1 d 2 Þ r 1 Þ. Similarly, L 2r G has also the same out-degrees, diameter k ¼ 2ðr þ 1Þþ1, and order N ¼ 2ððd 1 d 2 Þ rþ1 þðd 1 d 2 Þ r Þ. In particular, LG is a Moore bipartite digraph of diameter k ¼ 4. Furthermore, fl t Gg t is a family of dense bipartite digraphs with fixed out-degrees ðd 1 ; d 2 Þ and diameter kðtþ ¼3 þ t, since its order function NðtÞ satisfies that NðtÞ ¼ð1 "ÞMðd 1 ; d 2 ; 3 þ tþ, where " :¼ Mðd 1; d 2 ; 3 þ tþ NðtÞ Mðd 1 ; d 2 ; 3 þ tþ 8 ðd 1 d 2 Þ r 1 1 >< ðd ¼ 1 d 2 Þ rþ1 1 ðd 1 d 2 Þ r 1 >: ðd 1 d 2 Þ rþ2 1 if t ¼ 2r 1; if t ¼ 2r; is always smaller than 1=ðd 1 d 2 Þ 2.

5 ON MOORE BIPARTITE DIGRAPHS 175 Finally, in Section 3, we show that the line digraph structure appears as a characteristic and extremal property of the class of Moore bipartite digraphs of diameter four. That is, any such digraph is the line digraph of a Moore bipartite digraph of diameter three. As a result, the enumeration of Moore bipartite digraphs has been reduced to the case of diameter three. 2. MOORE BIPARTITE DIGRAPHS OF DIAMETER THREE From (6), the problem of enumeration of ðd 1 ; d 2 Þ-outregular Moore bipartite digraphs of diameter k ¼ 3 leads us to the search of all binary matrices A 1 ; A 2 of order n ¼ 1 þ d 1 d 2 satisfying that A J n ¼ d J n and A A ¼ J n I n : ð10þ Next, we present a characterization of these solutions, from which the semi-inregularity of their associated digraphs is derived. Lemma 2.1. Let A 1 2M n ð0; 1Þ be a d 1 -outregular matrix of order n ¼ 1 þ d 1 d 2. Then, there is a ðd 1 ; d 2 Þ-outregular Moore bipartite digraph G ¼ðV 1 [ V 2 ; EÞ of diameter k ¼ 3 such that A 1 represents the adjacencies of G from V 1 to V 2, if and only if, A 1 is nonsingular and d 1 -regular (that is, J n A 1 ¼ A 1 J n ¼ d 1 J n ) such that A 2 ¼ A 1 1 ðj n I n Þ is a (d 2 -regular) binary matrix. In this case, A 2 represents the adjacencies of G from V 2 to V 1. Proof. Assume first that G is a ðd 1 ; d 2 Þ-outregular Moore bipartite digraph of diameter k ¼ 3, such that A 1 is a submatrix of its adjacency matrix A ¼ bip ða 1 ; A 2 Þ. Then, since A 1 A 2 ¼ J n I n and detðj n I n Þ¼ ð 1Þ n 1 ðn 1Þ 6¼ 0, we deduce that A 1 is nonsingular and A 2 ¼ A 1 1 ðj n I n Þ2M n ð0; 1Þ. Moreover, using the identity A 2 A 1 ¼ J n I n, we obtain that ðj n I n ÞA 1 1 ¼ A 2 ¼ A 1 1 ðj n I n Þ, which implies that J n A 1 ¼ A 1 J n ¼ d 1 J n and A 1 is d 1 -regular. Conversely, if A 1 2M n ð0; 1Þ is a d 1 -regular non-singular matrix such that A 2 ¼ A 1 1 ðj n I n Þ is also a ð0; 1Þ-matrix, then A 1 A 2 ¼ J n I n ¼ A 2 A 1, since A 1 1 J n ¼ J n A 1 1. Moreover, from A 1J n ¼ d 1 J n and n ¼ 1 þ d 1 d 2, we get A 1 1 J n ¼ 1 d 1 J n ¼ d 2 n 1 J n, whence A 2 J n ¼ A 1 1 ðj n I n ÞJ n ¼ðn 1ÞA 1 1 J n ¼ d 2 J n : Consequently, the associated bipartite digraph with binary matrix A ¼ bip ða 1 ; A 2 Þ is a Moore bipartite digraph of diameter k ¼ 3. & The following property is a direct consequence of Lemma 1.1, taking into account that the column sums of the submatrix A represent the in-degrees of the vertices in V.

6 176 JOURNAL OF GRAPH THEORY Corollary 2.1. Let G be a ðd 1 ; d 2 Þ-outregular Moore bipartite digraph of diameter k ¼ 3. Then, G is ðd 2 ; d 1 Þ-inregular. In particular, G is regular if d 1 ¼ d 2. From the above, given the out-degrees ðd 1 ; d 2 Þ, we are interested in finding non-singular d 1 -regular matrices A 1 2M n ð0; 1Þ, where n ¼ 1 þ d 1 d 2, such that A 1 1 ðj n I n Þ is also a ð0; 1Þ-matrix. In this work, we confine our search to the class of g-circulant matrices. A well-known property of these matrices states that if A, ¼ 1; 2, are g -circulant matrices of order n, then A 1 A 2 is a ðg 1 g 2 Þ- circulant matrix and its Hall polynomial is A1 A 2 ðxþ ¼ A1 ðx g 2 Þ A2 ðxþ; where the polynomial arithmetic is modulo x n 1; see [1]. Moreover, if A is non-singular and has order n, then g has an inverse in Z n (g 2 Z n ) and A 1 is a ðg 1 Þ-circulant matrix; see [5]. Thus, for instance, since the corresponding A 1 matrix of the digraph BDðd 1 ; d 2 ; nþ is a non-singular d 1 -circulant matrix, with Hall polynomial A1 ðxþ ¼1 þ x þþx d1 1, it follows that A 1 1 is ðd1 1Þ- circulant, with d1 1 ¼ d 2, and has Hall polynomial A 1 ðxþ such that A 1 1 ðx n d 2 Þ A 1 ðxþ ¼1. Then, taking into account that J n I 1 n is 1-circulant, we have that A 2 ¼ A 1 1 ðj n I n Þ is a ð d 2 Þ-circulant matrix with Hall polynomial A2 ðxþ ¼ A 1 ðxþ J 1 n I n ðxþ ¼ 1 n d X2 1 x i þ 1 d 1 d 1 d 1 i¼0 X n 1 i¼n d 2 x i ¼ x n d 2 þ x n d 2þ1 þþx n 1 :! X n 1 x i i¼1 We point out that if A 1 is a g-circulant nonsingular matrix of order n, then there is a permutation matrix P such that A 0 1 :¼ PA 1 is a circulant matrix with the same Hall polynomial as A 1 ; see again [5]. Moreover, A 2 ¼ A 1 1 ðj n I n Þ is a ðg 1 Þ- circulant matrix and A 0 2 :¼ðA0 1 Þ 1 ðj n I n Þ is a circulant matrix such that A 0 2 ¼ A 1 1 P> ðj n I n Þ¼A 1 1 ðj n I n ÞP > ¼ A 2 P > ; where we have used that P 1 ¼ P > and PJ n ¼ J n P since P is a permutation matrix. Therefore, A ¼ bip ða 1 ; A 2 Þ and A 0 :¼ bip ða 0 1 ; A0 2 Þ¼bip ðpa 1; A 2 P > Þ are the adjacency matrices of isomorphic digraphs since A 0 ¼ 0 A0 1 A 0 ¼ P 0 > 0 A1 P 0 : ð11þ I A I In the particular case of the digraph BDðd 1 ; d 2 ; nþ, the permutation of the rows of A 1 (which is d 1 -circulant) giving a circulant matrix A 0 1 is represented by a

7 ð d 2 Þ-circulant matrix P (recall that d 1 1 ¼ d 2 ) with Hall polynomial P ðxþ ¼1, and we get A 0 1 ¼ PA 1. Then, A 0 2 ¼ A 2P > is also a circulant matrix with Hall polynomial A 0 2 ðxþ ¼ A2 ðx d 1 Þ P >ðxþ ¼x ðn d 2Þd 1 þ x ðn d 2þ1Þd 1 þþx ðn 1Þd 1 ¼ x þ x 1þd 1 þþx 1þðd 2 1Þd 1 : ON MOORE BIPARTITE DIGRAPHS 177 Hence, we can restrict our attention to the case of (ordinary) circulant matrices. In the following result, we reformulate the conditions on such matrices in two ways: First, in additive terms, by showing the presence of a near-factorization of the cyclic group Z n ; and second in terms of the coefficients of their Hall polynomials. Proposition 2.1. For some given positive integers d 1 ; d 2, let A 1 2M n ð0; 1Þ be a circulant matrix with Hall polynomial A1 ðxþ ¼x i 1 þ x i 2 þþx i d 1, where 0 i 1 < i 2 < < i d1 < n ¼ 1 þ d 1 d 2, and let S 1 ¼fi 1 ; i 2 ;...; i d1 g. Then, the following statements are equivalent: ðiþ There is a ð0; 1Þ-matrix A 2 such that A ¼ bip ða 1 ; A 2 Þ is the adjacency matrix of a ðd 1 ; d 2 Þ-outregular Moore bipartite digraph of diameter k ¼ 3; ðiiþ There exists a subset S 2 Z n of cardinal d 2 such that S 1 þ S 2 ¼ Z n nf0g; ð12þ ðiiiþ The polynomials A1 ðxþ and x n 1 are relatively prime and A1 ðxþ 1 is a polynomial with d 2 coefficients equal to ð1 d 1 Þ=d 1 and n d 2 coefficients equal to 1=d 1. Proof. First, we notice that A 1 is d 1 -regular, since A 1 2M n ð0; 1Þ is circulant and A1 ð1þ ¼d 1. Thus, from Lemma 1.1, and recalling that the inverse of a nonsingular matrix is also circulant, we can reformulate (i) by saying that there exists a circulant matrix A 2 2M n ð0; 1Þ such that A 1 A 2 ¼ J n I n or, equivalently, there is a polynomial A2 ðxþ ¼x j 1 þ x j 2 þþx j d 2, with 0 j 1 < j 2 < < j d2 < n, satisfying that A1 ðxþ A2 ðxþ ¼ X ði;jþ2s 1 S 2 x iþj ¼ x þ x 2 þþx n 1 ; where S 2 ¼fj 1 ; j 2 ;...; j d2 gz n. In additive terms, such a polynomial relation can be expressed as and, consequently, ðiþ ()ðiiþ. S 1 þ S 2 ¼fi þ j; ði; jþ 2S 1 S 2 g¼z n nf0g

8 178 JOURNAL OF GRAPH THEORY Besides, since a circulant matrix A 1 is nonsingular if, and only if, there exists the inverse polynomial of A1 ðxþ in Q½xŠ=ðx n 1Þ, it turns out that ðiþ is also equivalent to saying that A1 ðxþ and x n 1 are relatively prime and the polynomial A1 ðxþ 1 ¼ A 1 ðxþ ¼P n 1 1 i¼0 a ix i, with A 1 ð1þ ¼P n 1 1 i¼0 a i ¼ 1=d 1, satisfies A2 ðxþ ¼ A 1 1 ðxþ J n I n ðxþ ¼ Xn 1 i¼0 a i x i! X n 1 i¼1 x i! ¼ða 1 þ a 2 þþa n 1 Þþþða 0 þ a 1 þþa n 2 Þx n 1 ¼ 1 a 0 þ 1 a 1 x þþ 1 a n 1 x n 1 ; d 1 d 1 d 1 which means that d 2 coefficients of A 1 ðxþ are equal to ð1 d 1Þ=d 1 1 and the remaining ones are equal to 1=d 1, since A 2 ¼ A 1 1 ðj n I n Þ2M n ð0; 1Þ and A2 ð1þ ¼d 2. & From now on, we will use the additive version ðiiþ since it turns out to be more convenient from the theoretical point of view, whereas the polynomial version ðiiiþ seems to be more appropriate for a computational exploration. Thus, we want to find solutions to (12); that is, pairs ðs 1 ; S 2 Þ of subsets of Z n, n ¼ 1 þ d 1 d 2, with cardinality js j¼d, such that all the sums i þ j, where ði; jþ 2S 1 S 2, are distinct and non-null. In the context of group theory, this is a special type of (t-)near-factorization of Z n, where it is only required that js 0 1 þ S0 2 j¼js0 1 jjs0 2 j or, equivalently, S0 1 þ S0 2 ¼ Z nnftg for some t 2 Z n.ofcourse, every t-near-factorization provides us with a solution to our problem or 0-nearfactorization, by simply taking S 1 ¼ t þ S 0 1 and S 2 ¼ S 0 2. The pair of integer sets constituting a near-factorization are also known in the literature as number systems (see e.g., Grinstead [13]) because of their similarity to the standard number systems extensively studied. As we shall see later, an easy way of obtaining such systems is from the so-called degenerate British Number Systems (introduced by De Bruijn [4] in the context of the non-negative integers). Grinstead [13] realized that, if ðs 1 ; S 2 Þ is a number system, then ðs 1 þ s; S 2 þ tþ with s; t 2 Z n, ðhs 1 ; hs 2 þ tþ with h 2 Z n, and ð S 1; S 2 Þ also are number systems, and all of them are considered isomorphic to ðs 1 ; S 2 Þ. From the above, one can prove that every isomorphic class contains a near-factorization ðs 1 ; S 2 Þ; satisfying S ¼ S, which is called symmetric (see Caen et al. [6]). In the case of our 0-near-factorizations, however, this is not true because not all the above transformations fix the zero. Before concentrating in such a case, let us state the following general result, which was proved by Bacsó et al. in [3]: Lemma 2.2. Let ðs 1 ; S 2 Þ be a near-factorization of Z n with js 1 j2. Then, gcdðs 1 [fngþ ¼ 1: ð13þ

9 In our context, let us denote by S the set of all number systems corresponding to 0-near-factorizations S :¼ fðs 1 ; S 2 Þ2P d1 ðz n ÞP d2 ðz n Þ; S 1 þ S 2 ¼ Z n nf0gg; and by BDðS 1 ; S 2 Þ the Moore bipartite digraph associated with ðs 1 ; S 2 Þ2S, which means that the submatrices A 1 :¼ AðS 1 Þ and A 2 :¼ AðS 2 Þ of its adjacency matrix A are circulant with Hall polynomials A1 ðxþ ¼ P i2s 1 x i and A2 ðxþ ¼ P j2s 2 x j, respectively. In graphical terms, the adjacencies of the digraph BDðS 1 ; S 2 Þ, with set of vertices V ¼f1; 2gZ n, are ð; iþ!ð; i þ S Þ. Thus, using this notation BDðd 1 ; d 2 ; nþ ¼BDðf0; 1;...; d 1 1g; f1; 1 þ d 1 ;...; 1 þðd 2 1Þd 1 gþ: Now the set S is closed under some translations and homothetic transformations, in the sense that if ðs 1 ; S 2 Þ2S, then ðt þ S 1 ; t þ S 2 Þ2S, for each t 2 Z n and ðh S 1 ; h S 2 Þ2Sif h 2 Z n. In the following lemma, we show that, in order to construct non-isomorphic digraphs, it is necessary to consider the quotient set of S modulo such transformations. Lemma 2.3. Let ðs 1 ; S 2 Þ be a 0-near-factorization of Z n, where js j¼d and n ¼ 1 þ d 1 d 2. Let BDðS 1 ; S 2 Þ be its associated ðd 1 ; d 2 Þ-outregular Moore bipartite digraph of diameter three. Then, ðiþ BDðt þ S 1 ; t þ S 2 Þ is isomorphic to BDðS 1 ; S 2 Þ, for each t 2 Z n ; ðiiþ BDðh S 1 ; h S 2 Þ is isomorphic to BDðS 1 ; S 2 Þ, for each h 2 Z n ; ðiiiþ BDðS 1 ; S 2 Þ is isomorphic to BDðd 1 ; d 2 ; nþ if, for some, S is an arithmetic progression with common difference h 2 Z n modulo n. Proof. Let A ðtþ 1 :¼ Aðt þ S 1 Þ and A ð tþ 2 :¼ Að t þ S 2 Þ be the circulant submatrices associated with ðt þ S 1 ; t þ S 2 Þ2S. Then, A ðtþ 1 ¼ PA 1 and A ð tþ 2 ¼ A 2 P > ; where P is a permutation circulant matrix (with P ðxþ ¼x t ). Thus, taking into account (13), we deduce that all Moore bipartite digraphs BDðt þ S 1 ; t þ S 2 Þ are isomorphic. Analogously, if we denote by A ½hŠ :¼ Aðh S Þ the circulant submatrices associated with ðh S 1 ; h S 2 Þ2S, then! 0 A ½hŠ 1 ¼ A ½hŠ P > A 1 P P > A 2 P 0 ON MOORE BIPARTITE DIGRAPHS 179 > 0 A 1 P 0 A P ¼ P 0 0 P where P is a permutation h-circulant matrix ( P ðxþ ¼1), since h 2 Z n. Therefore, BDðh S 1 ; h S 2 Þ and BDðS 1 ; S 2 Þ are isomorphic. ;

10 180 JOURNAL OF GRAPH THEORY Finally, if S 1 is an arithmetic progression with common difference h 2 Z n modulo n; that is, S 1 ¼ t þ hf0; 1;...; d 1 1g, then S 2 ¼ tþhf1; 1 þ d 1 ; ; 1 þðd 2 1Þd 1 g and, consequently, from ðiþ and ðiiþ, we conclude that BDðS 1 ; S 2 ÞffiBDðd 1 ; d 2 ; nþ. & Notice that, in graphical terms, the isomorphism from the vertex set of BDðS 1 ; S 2 Þ to the vertex set of BDðt þ S 1 ; t þ S 2 Þ is just ð1; iþ 7!ð1; i tþ and ð2; iþ 7!ð2; iþ. Correspondingly, the isomorphism from BDðS 1 ; S 2 Þ to BD ðh S 1 ; h S 2 Þ is ð; iþ 7!ð; hiþ. Intuitively, we can imagine a pair ðs 1 ; S 2 Þ2S as a kind of tiling of Z n nf0g, where S 1 represents the set of initial positions of the 1 1 tiles and S 2 provides the translation steps such that ft þ S 1 g t2s2 completely fills Z n nf0g. When we employ the usual integer arithmetic, this approach suggests the construction of a solution to S 1 þ S 2 ¼f1;...; d 1 d 2 g, where d ¼, by composing a (simpler) solution to R 1 þ R 2 ¼f1;...; 1 2 g with one to T 1 þ T 2 ¼f1;...; 1 2 g. The idea is to blow up each unit tile centered at R 1 þ R 2 to transform it into all the tiles of T 1 þ T 2. In fact, this leads to a special type of what is called a degenerate British Number System [13], or De Bruijn nearfactorization of the cyclic group Z n [3]. The second name is because such factorizations are the discrete version of a construction of De Bruijn, which gives all the possible number systems for the nonnegative integers (see [4] for more details). As we will see later, the interest of our De Bruijn (0-)near-factorizations is that, contrarily to the case when they are used to construct graphs [13,3], they sometimes give non-isomorphic bipartite digraphs. We recall that in the following result we use standard integer arithmetic. Lemma 2.4. Let ðr 1 ; R 2 Þ and ðt 1 ; T 2 Þ be two pairs of subsets of nonnegative integers such that R 1 þ R 2 ¼f1;...; 1 2 g and T 1 þ T 2 ¼f1;...; 1 2 g, with jr j¼ and jt j¼. If 0 2 T 1, then ðr 1 þ 1 2 T 1 ; R 2 þ 1 2 ðt 2 1ÞÞ is a solution to S 1 þ S 2 ¼f1;...; d 1 d 2 g, where js j¼d ¼. Proof. Notice that each of the elements f0; 1g belongs to exactly one of the integer subsets T, since 1 2 T 1 þ T 2 and jt 1 \ T 2 j1, which means that minðt 2 1Þ ¼0, if 0 2 T 1. Clearly, S 1 þ S 2 ¼ðR 1 þ 1 2 T 1 ÞþðR 2 þ 1 2 ðt 2 1ÞÞ ¼ R 1 þ R 2 þ 1 2 ðt 1 þ T 2 1Þ ¼f1;...; 1 2 gþf0;...; ð 1 2 1Þ 1 2 g ¼f1;...; d 1 d 2 g; where js j¼jr jjt j. &

11 ON MOORE BIPARTITE DIGRAPHS 181 Thus, for instance, in the regular even case d 1 ¼ d 2 ¼ 2, we can compose the pair ðr 1 ; R 2 Þ, R 1 ¼f0; 1;...; 1g and R 2 ¼f1; 1 þ ;...; 1 þð 1Þg; with the unique solution to T 1 þ T 2 ¼ Z 5 nf0g (modulo transformations of the type T 7! t þ ht), T 1 ¼f0; 1g and T 2 ¼f1; 3g; in two different ways: ðm1þ S 1 ¼ R 1 þ 2 T 1 ¼f0; 1;...; 1; 2 ; 2 þ 1;...; 2 þ 1g S 2 ¼ R 2 þ 2 ðt 2 1Þ ¼f1; 1 þ ;...; 2 þ 1; 2 2 þ 1;...; 3 2 þ 1g ðm2þ S 1 ¼ R 2 þ 2 T 1 ¼f1; 1 þ ;...; 2 þ 1; 2 þ 1;...; 2 2 þ 1g S 2 ¼ R 1 þ 2 ðt 2 1Þ ¼f0; 1;...; 1; 2 2 ; 2 2 þ 1;...; 2 2 þ 1g: While the Moore bipartite digraph obtained from (M2) is isomorphic to BDð2; 1 þ 4 2 Þ, since the corresponding set S 1 ¼ 1 þ f0; 1;...; 2 1g is an arithmetic progression with common difference 2 Z 1þ42, the associated digraph with the pair defined by (M1) is a new construction, as we will show in the next theorem that generalizes the previous ideas. Before this, let us give a particular example: For d ¼ 4(¼2), we get ðm1þ S 1 ¼f0; 1; 4; 5g S 2 ¼f1; 3; 9; 11g; ðm2þ S 1 ¼f1; 3; 5; 7g S 2 ¼f0; 1; 8; 9g; from which we construct two nonisomorphic regular Moore bipartite digraphs of diameter k ¼ 3 and degree d ¼ 4, which turn out to be unique if we just consider digraphs of the type BDðS 1 ; S 2 Þ. Theorem 2.1. Let d 1 d 2 be positive integers. Given a factorization, d 1 ¼ 1 1 and d 2 ¼ 2 2 ; where 1 ð ¼ 1; 2Þ;

12 182 JOURNAL OF GRAPH THEORY let us consider the pair ðs 1 ; S 2 Þ of subsets of Z n, where n ¼ 1 þ d 1 d 2, defined by S 1 ¼[ 1 1 i¼0 ðr 1 þ i 1 2 Þ and S 2 ¼[ 2 1 j¼0 ðr 2 þ j Þ; ð14þ where R 1 ¼f0; 1;...; 1 1g and R 2 ¼f1; 1 þ 1 ;...; 1 þð 2 1Þ 1 g. Then, the Moore bipartite digraph BDðS 1 ; S 2 Þ is not isomorphic to BDðd 1 ; d 2 ; nþ, if and only if, > 1. Proof. If we take the integer subsets, T 1 ¼f0; 1;...; 1 1g and T 2 ¼f1; 1 þ 1 ;...; 1 þð 2 1Þ 1 g; then S 1 ¼ R 1 þ 1 2 T 1 and S 2 ¼ R 2 þ 1 2 ðt 2 1Þ. So, applying Lemma 2.4, we have that S 1 þ S 2 ¼f1;...; d 1 d 2 g, where js j¼d. We know that if G ¼ðV 1 [ V 2 ; EÞ is a Moore bipartite digraph of diameter three, then given a vertex v 1 2 V 1 there is a unique v 1! w walk of length two, for each w 2 V 1 nf0g. Moreover, since d ðv 2 Þ¼d 1, for each v 2 2 V 2, it follows that there are exactly d 1 ðd 1 1Þ arcs incident from vertices of N2 þðv 1Þ¼V 1 nfv 1 g (at distance two from v 1 ) to vertices of N þ ðv 1 Þ. We will show that these arcs are distributed in a different manner when we consider G ¼ BDðd 1 ; d 2 ; nþ and G ¼ BDðS 1 ; S 2 Þ,if > 1. First, we point out that their automorphisms group act transitively on each partite set V, since the associated submatrix A is circulant. Thus, let us take the vertex v 1 ¼ 0ofV 1 and let us compute the maximum number of vertices of N þ ð0þ that are adjacent from a given vertex of N2 þ ð0þ ¼f1; 2;...; n 1g. In the case of the digraph BDðd 1 ; d 2 ; nþ, such a value is maxfjn þ ðwþ\n þ ð0þj; w 2 N þ 2 ð0þg ¼ d 1 1; which is attained by the vertices 1 and n 1, since N þ ð0þ ¼f0; 1;...; d 1 1g and A 1 is circulant. Nevertheless, in the case of the digraph BDðS 1 ; S 2 Þ such a maximum is d 1 minf 1 ; 1 g¼d 1 1 < d 1 1, since there is a partition of N þ ð0þ into 1 > 1 non-contiguous segments R 1 þ i 1 2 ¼fi 1 2 ;...; i 1 2 þ 1 1g of cardinality 1, where the greatest integer of N þ ð0þ is 1 1þ 1 2 ð 1 1Þ < ðn 1Þ=2 (since 2 > 1). Hence, there does not exist an isomorphism between both digraphs that identify the vertices v ¼ 0 of the partite set V 1 of each digraph, which implies that they are not isomorphic, if d 1 > d 2. This also holds in the regular case d :¼ d 1 ¼ d 2, since BDðd; 1 þ d 2 Þ is vertextransitive; see [9]. Notice that in the extremal case 1 ¼ 1or 2 ¼ 1, the digraph BDðS 1 ; S 2 Þ is isomorphic to BDðd 1 ; d 2 ; nþ, since S 1 (respectively, S 2 ) represents an arithmetic progression modulo n ¼ 1 þ d 1 d 2 with common difference s ¼ 1 (respectively, s ¼ 1 2 Z n ); see Lemma 2.3. &

13 ON MOORE BIPARTITE DIGRAPHS 183 As a consequence, if we have two distinct factorizations of the composite degrees d ¼ ¼ 0 0, where > 1 and 0 0 > 1, such that 1 6¼ 0 1 ; if d 1 > d 2 ; and f 1 ; 2 g 6¼ f 0 1 ;0 2 g; if d 1 ¼ d 2 ; then BDðS 1 ; S 2 Þ and BDðS 0 1 ; S0 2Þ are not isomorphic. Hence, the previous theorem provides new constructions of Moore bipartite digraphs with composite outdegrees ðd 1 ; d 2 Þ, where d 1 d 2 > 1. Notice that if d 2 ¼ 1, there is a unique solution to our enumeration problem, namely BDðd 1 ; 1; 1 þ d 1 Þ. In the cases d 1 ¼ 2; 3, we have also showed that the only possible Moore bipartite digraphs are the BDðd 1 ; d 2 ; d 1 d 2 þ 1Þ. Here we only discuss the simple case d 1 ¼ 2: By Lemma 2.3 (i) we can assume that S 1 ¼ f0; sg, so that the isomorphism BDðS 1 ; S 2 ÞffiBDð2; d 2 ; 2d 2 þ 1Þ is a direct consequence of Lemma 2.2 and Lemma 2.3 (iii). 3. MOORE BIPARTITE DIGRAPHS OF DIAMETER FOUR As we have already mentioned, the line digraph LG of a Moore bipartite digraph G with diameter three and out-degrees ðd 1 ; d 2 Þ, where d 1 d 2 > 1, is a Moore bipartite digraph of diameter four. Next, we will prove that the converse result is also true and, therefore, the problem of the enumeration of Moore bipartite digraphs can be reduced to the case of diameter three. In fact, it is shown that the rank of the adjacency matrix A of a ðd 1 ; d 2 Þ- outregular Moore bipartite digraph of diameter four attains the minimum possible value, rank A ¼ 2ð1 þ d 1 d 2 Þ, whence the line digraph structure follows. This approach has already been used in some other ð0; 1Þ-matrix equations; see [17,10]. First, we will see that all Moore bipartite digraphs with a given diameter k 2f3; 4g and fixed out-degrees ðd 1 ; d 2 Þ are cospectral. Lemma 3.1. Let G be a ðd 1 ; d 2 Þ-outregular Moore bipartite digraph of diameter k > 2. Then, the spectrum of G is pffiffiffiffiffiffiffiffiffi p d 1 d 2 i i ffiffiffiffiffiffiffiffiffi! d 1 d 2 1 d 1 d 2 d 1 d 2 1 pffiffiffiffiffiffiffiffiffi p d 1 d 2 i 0 i ffiffiffiffiffiffiffiffiffi! d 1 d 2 1 d 1 d 2 ðd 1 þ d 2 2Þð1 þ d 1 d 2 Þ d 1 d 2 1 (where i ¼ p ffiffiffiffiffiffi 1 ). if k ¼ 3; if k ¼ 4;

14 184 JOURNAL OF GRAPH THEORY Proof. Let A ¼ bip ða 1 ; A 2 Þ be the adjacency matrix of G ¼ðV 1 [ V 2 ; EÞ and let n ¼jV j. From the out-regularity of A (that can also be expressed as A j ¼ d j, where j is the all-one vector), it turns out that p ffiffiffiffi pffiffiffiffi p ffiffiffiffi p ffiffiffiffi p ffiffiffiffi pffiffiffiffi ð d 1; j; d 2 jþ :¼ ð d 1;... n 1 ; d 1; d 2 ;... n 2 ; d 2Þ pffiffiffiffiffiffiffiffiffi is a positive eigenvector of A associated to the eigenvalue d 1 d 2. Since A is irreducible p (G ffiffiffiffiffiffiffiffiffi is strongly connected) and using Perron-Frobenius theorem, we know that d 1 d 2 has multiplicity equal to 1. Moreover, since G is bipartite, the characteristic polynomial of G, ðg; xþ, satisfies that ðg; xþ ¼ ð 1Þ n 1þn 2 p ðg; xþ, which implies that ffiffiffiffiffiffiffiffiffi d 1 d 2 is also a simple eigenvalue of G; see e.g. [11,16]. Furthermore, from the matrix equation (4) of the case k ¼ 3, I þ A þ A 2 þ A 3 ¼ J n d 1 J n ; ð15þ d 2 J n J n where n :¼ n ¼ 1 þ d 1 d 2, and taking into account that 0 is an eigenvalue of the right-hand matrix of (17) with multiplicity equal to 2n 2, we deduce that the remaining 2n 2 eigenvalues of A must be roots of the equation 1 þ x þ x 2 þ x 3 ¼ðx þ 1Þðx 2 þ 1Þ ¼0. Hence, since ðg; xþ ¼ ðg; xþ, we have that i are eigenvalues of G with the same multiplicity n 1 ¼ d 1 d 2. Analogously, from the corresponding matrix equation (5) for k ¼ 4 A þ A 2 þ A 3 þ A 4 ¼ d 1J n1 J n1 ;n 2 J n2 ;n 1 d 2 J n2 where n ¼ d ð1 þ d 1 d 2 Þ, we obtain that i and 0 are eigenvalues of A with multiplicity m A ðiþ ¼m A ð iþ and m A ð0þ ¼n 1 þ n 2 2 2m A ðiþ, respectively. Therefore, using the condition tr A 2 ¼ 2d 1 d 2 2m A ðiþ ¼0, we conclude that m A ðiþ ¼d 1 d 2 and m A ð0þ ¼ðd 1 þ d 2 2Þð1 þ d 1 d 2 Þ. & ; Theorem 3.1. Every Moore bipartite digraph of diameter four is the line digraph of a Moore bipartite digraph of diameter three. Proof. Let A ¼ bip ða 1 ; A 2 Þ be the adjacency matrix of a ðd 1 ; d 2 Þ-outregular Moore bipartite digraph G ¼ðV 1 [ V 2 ; EÞ of diameter k ¼ 4. From (7), we know that A 1 ði n2 þ A 2 A 1 Þ¼J n1 ;n 2 and A 2 ði n1 þ A 1 A 2 Þ¼J n2 ;n 1 ; ð16þ

15 where n ¼jV j¼d ð1 þ d 1 d 2 Þ. Thus, B :¼ A þ A 3 ¼ 0 A 1 ði n2 þ A 2 A 1 Þ 0 J ¼ n1 ;n 2 A 2 ði n1 þ A 1 A 2 Þ 0 J n2 ;n 1 0 which corresponds to the adjacency matrix of the complete bipartite digraph K n1 ;n 2. Then, since ðb þ ffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi p n n 1 n 2 I n1 þn 2 ÞB ¼ 2 J n1 n 1 n 2 J n1 ;n pffiffiffiffiffiffiffiffiffi 2 n 1 n 2 n 1 J n2 J n2 ;n 1 and taking into account that A J ¼ d J, we deduce that p ða ffiffiffiffiffiffiffiffiffi d 1 d 2In1 þn 2 ÞðA þ A 3 p þ ffiffiffiffiffiffiffiffiffi n 1 n 2 I n1 þn 2 ÞðA þ A 3 Þ¼0; which implies that the minimum polynomial of A, A ðxþ, divides p qðxþ ¼ðx ffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi p d 1 d 2Þðx þ d 1 d 2Þðx 2 ffiffiffiffiffiffiffiffiffi d 1 d 2x þ 1 þ d1 d 2 Þðx 2 þ 1Þx: But, since A ðxþ also divides the characteristic polynomial of G, which is ðg; xþ ¼ðx 2 d 1 d 2 Þðx 2 þ 1Þ d 1d 2 x n 1þn 2 2ð1þd 1 d 2 Þ (by Lemma 3.1), we conclude that Hence, A ðxþ ¼ðx 2 d 1 d 2 Þðx 2 þ 1Þx: rank A ¼ n 1 þ n 2 dim ker A ¼ n 1 þ n 2 m A ð0þ ¼2ð1 þ d 1 d 2 Þ: Taking into account that rank A ¼ rank A 1 þ rank A 2 and minfrank A 1 ; rank A 2 gminfn 2 =d 1 ; n 1 =d 2 g¼1 þ d 1 d 2 ; since A J ¼ d J and each column of A is non-null, it turns out that rank A 1 ¼ rank A 2 ¼ 1 þ d 1 d 2 : ON MOORE BIPARTITE DIGRAPHS 185 This extremal property implies that the rows of A are mutually orthogonal or identical, which is equivalent, by Heuchenne s condition [15], to say that G is a line digraph. Therefore, we conclude that G ¼ LG 0, where G 0 is a Moore bipartite digraph with diameter three and out-degrees ðd 1 ; d 2 Þ. & ;

16 186 JOURNAL OF GRAPH THEORY As a consequence of the previous theorem and Corollary 2.1, we deduce that any Moore bipartite digraph G ¼ðV 1 [ V 2 ; EÞ of diameter four is semidiregular, which means that d þ ðvþ ¼d ðvþ ¼d ; 8v 2 V, since each vertex v of G ¼ LG 0 represents an arc ðu; wþ, say, of G 0 and the corresponding degrees satisfy the relation d G ðvþ ¼d G 0ðuÞ ¼dþ G 0ðwÞ ¼dþ G ðvþ: Finally, as a result of Theorem 2.1, and taking into account that the line digraph transformation is one-to-one from the class of strongly connected digraphs onto the class of line digraphs, up to isomorphisms (see [14]), we obtain new constructions of Moore bipartite digraphs of diameter four and composite degrees. Lemma 3.2. Let d 1 d 2 be positive integers such that d 1 d 2 > 1. Let d 1 ¼ 1 1 and d 2 ¼ 2 2, with 1, be integer factorizations. If ðs 1 ; S 2 Þ represents the solution to S 1 þ S 2 ¼ Z 1þd1 d 2 nf0g defined by (14), then L BDðS 1 ; S 2 Þ is a Moore bipartite digraph of diameter k ¼ 4 and degrees ðd 1 ; d 2 Þ, which is not isomorphic to L BDðd 1 ; d 2 ; 1 þ d 1 d 2 Þ,if > 1. REFERENCES [1] C. M. Ablow and J. L. Brenner, Roots and canonical forms for circulant matrices, Trans Amer Math Soc 107 (1963), [2] M. Aïder, Réseaux d interconnexion bipartis orientés (French English, French summary) Rev Maghrébine Math 1 (1992), no. 1, [3] G. Bacsó, E. Boros, V. Gurvich, F. Maffray, and M. Preissmann, On minimal imperfect graphs with circular symmetry, J Graph Theory 29 (1998), no. 4, [4] N. G. de Bruijn, On number systems, Nieuw Arch Wisk (3) 4 (1956), [5] P. J. Davis, Circulant Matrices, Wiley-Interscience, New York, [6] D. de Caen, D. A. Gregory, I. G. Hughes, and D. L. Kreher, Near-factors of finite groups, Ars Combin 29 (1990), [7] M. A. Fiol, I. Alegre, and J. L. A. Yebra, Line digraphs iterations and the ðd; kþ problem for directed graphs, in: Proc 10th Int Symp Comput Arch, Stockholm, 1983, pp [8] M. A. Fiol, J. L. A. Yebra, and I. Alegre, Line digraphs iterations and the ðd; kþ digraph problem, IEEE Trans Comput C-32 (1984), [9] M. A. Fiol and J. L. A. Yebra, Dense bipartite digraphs, J Graph Theory 14 (1990), [10] J. Gimbert and Y. Wu, The underlying line digraph structure of some ð0; 1Þmatrix equations, Discrete Appl Math 116 (2002),

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