Optimal extensions and quotients of 2--Cayley Digraphs
OOptimal extensions and quotientsof 2–Cayley Digraphs ∗ F. Aguil´o, A. Miralles and M. Zaragoz´a
Departament de Matem`atica Aplicada IVUniversitat Polit`ecnica de CatalunyaJordi Girona 1-3 , M`odul C3, Campus Nord08034 Barcelona.
October 8, 2018
Abstract
Given a finite Abelian group G and a generator subset A ⊂ G of cardinality two, weconsider the Cayley digraph Γ = Cay(G , A ). This digraph is called 2–Cayley digraph. An extension of Γ is a 2–Cayley digraph, Γ (cid:48) = Cay(G (cid:48) , A ) with G < G (cid:48) , such that there issome subgroup H < G (cid:48) satisfying the digraph isomorphism Cay(G (cid:48) / H , A ) ∼ = Cay(G , A ).We also call the digraph Γ a quotient of Γ (cid:48) . Notice that the generator set does not change.A 2–Cayley digraph is called optimal when its diameter is optimal with respect to itsorder.In this work we define two procedures, E and Q , which generate a particular typeof extensions and quotients of 2–Cayley digraphs, respectively. These procedures areused to obtain optimal quotients and extensions. Quotients obtained by procedure Q ofoptimal 2–Cayley digraphs are proved to be also optimal. The number of tight extensions,generated by procedure E from a given tight digraph, is characterized. Tight digraphs forwhich procedure E gives infinite tight extensions are also characterized. Finally, these twoprocedures allow the obtention of new optimal families of 2–Cayley digraphs and also theimprovement of the diameter of many proposals in the literature. Keywords : Cayley digraph, diameter, digraph isomorphism, minimum distance diagram,quotient, extension.
AMS subject classifications
Let G N be a finite Abelian group of order N generated by A = { a, b } ⊂ G N \ { } . TheCayley digraph Γ = Cay(G N , A ) is a directed graph with set of vertices V = G N and set of ∗ Research supported by the “Ministerio de Educaci´on y Ciencia” (Spain) with the European RegionalDevelopment Fund under projects MTM2011-28800-C02-01 and by the Catalan Research Council under project2014SGR1147.Emails: [email protected] , [email protected] , [email protected] a r X i v : . [ m a t h . C O ] M a y rcs E = { g → g + a, g → g + b : g ∈ G N } . These digraphs are called 2–Cayley digraphs.The concepts of ( directed ) path , distance , minimum path and diameter are the usual ones.We denote the diameter of Γ by D(G N , A ). Definition 1
Fixed N ≥ , the functions D and D are defined as D ( N ) = min { D(G N , A ) : G N cyclic , A ⊂ G N } and, for non square-free N , we also define D ( N ) = min { D(G N , A ) : G N non-cyclic , A ⊂ G N } . Most known proposals of 2–Cayley digraphs are given in terms of D -optimality and some-times, even more restricted, with generator set A = { , b } ⊂ Z N . However, D -optimalityalso has to be taken into account for non square-free order N . In this work the optimalitymeans D ( N ) = (cid:40) D ( N ) if N is square-free,min { D ( N ) , D ( N ) } otherwise . Table 1 shows D ( N ) and D ( N ) for several values of non square-free N . Notice differentbehaviors in the table, i.e. D < D , D = D and D > D , for some values of N . N lb( N ) D ( N ) Optimal Cyclic D ( N ) Optimal Non-cyclic8 3 3 Cay( Z , { , } ) 4 Cay( Z ⊕ Z , { (0 , , (1 , } )9 4 4 Cay( Z , { , } ) 4 Cay( Z ⊕ Z , { (0 , , (1 , } )12 4 5 Cay( Z , { , } ) 4 Cay( Z ⊕ Z , { (0 , , (1 , } )16 5 5 Cay( Z , { , } ) 6 Cay( Z ⊕ Z , { (0 , , (1 , } )18 6 6 Cay( Z , { , } ) 7 Cay( Z ⊕ Z , { (0 , , (1 , } )20 6 7 Cay( Z , { , } ) 6 Cay( Z ⊕ Z , { (0 , , (1 , } )Table 1: Some optimal 2-Cayley digraphs for non square-free orderMetrical properties of 2–Cayley digraphs can be studied using minimum distance diagrams (MDD for short). Sabariego and Santos [11] gave the algebraic definition of MDD in thegeneral case (for any cardinality of A ). Here we particularize their definition to 2-Cayleydigraphs. Definition 2 A minimum distance diagram related to the digraph Cay(G N , { a, b } ) is a map ψ : G N −→ N with the following two properties(a) for each η ∈ G N , ψ ( η ) = ( i, j ) satisfies ia + jb = η and (cid:107) ψ ( η ) (cid:107) is minimum among allvectors in N satisfying that property ( (cid:107) ( i, j ) (cid:107) = i + j ),(b) for every η ∈ G N and for every vector ( s, t ) ∈ N that is coordinate-wise smaller than ψ ( η ) , we have ( s, t ) = ψ ( γ ) for some γ ∈ G N (with sa + tb = γ ). These diagrams are also known as L -shapes when | A | = 2. They were used first by Wongand Coppersmith [13] in 1974 for cyclic groups and generator set of type { , s } . Fiol, Yebra,Alegre and Valero [7] in 1987 used L-shapes and their related tessellations to obtain infinitefamilies of tight 2–Cayley digraphs for cyclic groups, known as double-loop networks . There2re two complete surveys on double-loop networks, i.e. Bermond, Comellas and Hsu [2] in1995 and Hwang [8] in 2000.Minimum distance diagrams are usually represented by the image ψ (G N ), where each vector ψ ( η ) = ( i, j ) is depicted as a unit square [[ i, j ]] = [ i, i + 1] × [ j, j + 1] ∈ R (for every η ∈ G N ).A square [[ i, j ]] is labeled with the element ia + jb ∈ G N . L-shapes are usually denoted by thelengths of their sides, i.e. L = L( l, h, w, y ) with 0 ≤ w < l , 0 ≤ y < h and lh − wy = N . Theplane tessellation using the tile L is given by translation through the vectors u = ( l, − y ) and v = ( − w, h ) (see [7] for more details). Metrical properties of the digraph are contained in theirrelated diagrams. For instance, the distance between vertices in the digraph can be computedfrom any related MDD L . In particular, the diameter of the digraph Cay(G N , { a, b } ), denotedby D(G N , { a, b } ), is obtained from the so called diameter of L d L = l + h − min { w, y } − . (1)Fixed the area of L , N , a tight lower bound lb( N ) is known for d L (see for instance [7, 2]),lb( N ) = (cid:100)√ N (cid:101) − . (2)The value lb( N ) is also a tight lower bound for D ( N ).
05 27 49 611 8131015 121 143 (0 , , , ,
6) (0 , , , ,
7) (0 , , , ,
8) (0 , , , ,
9) (0 , , , ,
10) (0 , , , ,
11) (0 , , , ,
0) (0 , , , , , , , ,
2) (0 , , , ,
3) (0 , , , ,
4) (0 , , , , Figure 1: H F = L(5 , , ,
2) and H G = L(8 , , , , , ,
2) related to Cay( Z , { , } ) andL(8 , , ,
4) associated with Cay( Z ⊕ Z , { (0 , , (3 , } ). In Table 1, either D or D attain thelower bound. The first non square-free value of N with D ( N ) , D ( N ) > lb( N ) is N = 25,that is D (25) = D( Z , { , } ) = 8 and D (25) = D( Z ⊕ Z , { (0 , , (1 , } ) = 8 whilstlb(25) = 7. Definition 3
The digraph
Cay(G N , { a, b } ) is k –tight if D(G N , { a, b } ) = lb( N ) + k . Given a minimum distance diagram, H = L( l, h, w, y ), we also say (by analogy with its relateddigraph) that H is k –tight when d H = lb( lh − wy ) + k . According to these definitions, we say3he digraph Cay( Z , { , } ) is 0-tight (optimal) and Cay( Z ⊕ Z , { (0 , , (1 , } ) is 1-tight.0-tight digraphs are called tight (optimal) ones. There are optimal digraphs that are nottight, for instance Cay( Z ⊕ Z , { (0 , , (1 , } ) is 1-tight optimal. The following theoremgeometrically characterizes minimum distance diagrams. Theorem 1 ([1, Theorem 1]) H = L( l, h, w, y ) is a minimum distance diagram related tothe digraph Cay(G N , { a, b } ) if and only if lh − wy = N , la = yb and hb = wa in G N , ( l − y )( h − w ) ≥ and both factors do not vanish at the same time. Given a minimum distance diagram H = L( l, h, w, y ), we can find a 2–Cayley digraph associ-ated with H . The details can be found in [7, 5, 6] using the Smith normal norm , S , of the inte-gral matrix M = M ( l, h, w, y ) = (cid:18) l − w − y h (cid:19) . That is, S = diag( s , s ), s = gcd( l, h, w, y ), s | s , s s = N and S = U M V for some unimodular matrices
U, V ∈ Z × . More precisely,if U = (cid:18) u u u u (cid:19) then H is related to Cay( Z s ⊕ Z s , { ( u , u ) , ( u , u ) } ). Thus, if H isrelated to Cay(G N , { a, b } ), the group G N is cyclic if and only if gcd( l, h, w, y ) = 1. Althoughthis result is known since time ago, few authors have used it for 2–Cayley digraphs related tonon-cyclic groups. Clearly, for some non square-free values of N , non-cyclic groups are betterthan cyclic ones, as in Table 1.The motivation of this work appears from some numerical evidences associated with minimumdistance diagrams. Here we give four examples to remark some structural and metric detailsof 2–Cayley digraphs. Examples 1 and 2 are related to quotients whilst examples 3 and 4correspond to extensions. Quotients and extensions will be defined in the next section, butnow we want to highlight some numerical details using these examples.The definition of Cayley digraph isomorphism is the usual one, that is Γ = Cay(G , A ) ∼ =∆ = Cay(G , A ) whenever there is some isomorphism of groups f : G −→ G such thatthere is an arc g → g in Γ if and only if there is an arc f ( g ) → f ( g ) in ∆. Example 1
Let us consider
Γ = Cay( Z , { , } ) with related minimum distance diagram H =L(5 , , , . See the left hand side of Figure 1. The digraph Γ is tight with diameter D(Γ) = d H =lb(16) = 5 . Taking the subgroup H = { , } < Z , we get Γ (cid:48) = Cay( Z / H , { , } ) . Notice that H contains the minimum distance diagram H (cid:48) = L(4 , , , related to Γ (cid:48) ∼ = Cay( Z , { , } ) (labels and correspond to and modulo ). The diameter D(Γ (cid:48) ) = d H (cid:48) = 4 is not optimal since D( Z , { , } ) = D (8) = 3 = lb(8) . Example 1 shows a non-optimal quotient from an optimal 2–Cayley digraph. Notice how thealgebraic structure of this quotient is reflected in the tessellation of the MDD L(4 , , , Z , { , } ), through u = (4 ,
0) and v = ( − ,
2) with respect the tessellationof the MDD L(5 , , , Z , { , } ), through u (cid:48) = (5 , −
2) = u − v and v (cid:48) = ( − ,
4) = 2 v . Here, it is not clear how to obtain L(4 , , ,
0) from L(5 , , ,
2) withoutlooking at the lateral classes of H in Z . Example 2
Let us consider now
Γ = Cay( Z ⊕ Z , { (0 , , (3 , } ) , with minimum distance diagram H = L(8 , , , . See the right hand side of Figure 1. Γ is tight since D(Γ) = d H = lb(48) =10 . Here we take the subgroup H = { (0 , , (2 , } < Z ⊕ Z . Then, the quotient Cay( Z ⊕ Z / H , { (0 , , (3 , } ) ∼ = Cay( Z ⊕ Z , { (0 , , (3 , } ) = Γ (cid:48) has related minimum distance diagram H (cid:48) = L(4 , , , . This quotient Γ (cid:48) is also tight since D (Γ (cid:48) ) = d H (cid:48) = lb(12) = 4 . (cid:48) from an optimal digraph Γ. We can see how thetessellation by H is compatible with tessellation by H (cid:48) . In this example, unlike Example 1,it is clear how L(4 , , ,
2) is obtained from L(8 , , , Example 3
Consider the digraph
Γ = Cay( Z ⊕ Z , { (1 , , (0 , } ) with optimal diameter D(Γ) =4 = lb(9) and related minimum distance diagram H = L(3 , , , . The digraph Γ (cid:48) = Cay( Z ⊕ Z , { (1 , , (0 , } ) has related minimum distance diagram H (cid:48) = L(6 , , , . Γ (cid:48) is an extension of Γ taking the subgroup H = { (0 , , (3 , } < Z ⊕ Z . The diameter D(Γ (cid:48) ) = 10 is not optimal since D( Z , { , } ) = 9 = lb(36) . Example 4
Let us consider the tight digraph
Γ = Cay( Z , { , } ) ∼ = Cay( Z ⊕ Z , { (0 , , (1 , } ) with diameter D(Γ) = 4 = lb(11) and related minimum distance diagram H = L(4 , , , . Thedigraphs Γ m = Cay( Z m ⊕ Z m , { (0 , , (1 , } ) are extensions of Γ , with related minimum distancediagram H m = L(4 m, m, m, m ) , for m ≥ . Numerical calculations give D(Γ m ) = lb(11 m ) = 6 m − for m = 2 , . Thus, Γ and Γ are optimal extensions of Γ . Examples 3 and 4 show that extensions of optimal digraphs can be optimal or not.The previous examples allow us to define quotients and extensions of 2–Cayley digraphs fromtheir related minimum distance diagrams. From a given MDD H = L( l, h, w, y ) of area N = lh − wy , we can consider the L-shape m H = L( ml, mh, mw, my ) of area m N thatcorresponds to an extension-like procedure on the related digraphs. The same observationssuggest a quotient-like procedure from the related minimum distance diagram. By Theorem 2,the L-shape m H is also a minimum distance diagram.In Section 2 we define two procedures, for quotients and extensions, based on minimumdistance diagrams. Metrical properties of these procedures are also studied in that section.Theorem 3 shows that these kind of quotients are well suited from the metrical point of view.That is, quotients on optimal non-cyclic 2–Cayley digraphs are also optimal.Properties of these kind of extensions are studied in Section 3 and tight extensions are char-acterized. Tight digraphs with infinite tight extensions are proved to be (Lemma 3) thosehaving order 3 t , for some t ≥
1. Theorem 4 shows that these digraphs are always extensionsof the same digraph Cay( Z , { , } ). Thus, tight 2–Cayley digraphs with order N (cid:54) = 3 t cannot have infinite tight extensions of this type. Theorem 5 gives the exact number of tightextensions a tight digraph of order N can have. This number is called the extension coefficient c( N ). Proposition 3 shows that c( N ) can not be larger than O ( √ N ). Theorem 6 gives infinitefamilies of digraphs having maximum value of the extension coefficient, i.e. c( N ) = O ( √ N ).Finally, in Section 4, quotients and extensions of 2–Cayley digraphs are used to improve thediameter of some proposals in the literature. 5 Quotients and extensions of –Cayley digraphs Let us denote the non-negative integers by N . Given a minimum distance diagram H =L( l, h, w, y ) and m ∈ N , m (cid:54) = 0, we use the notation gcd( H ) = gcd( l, h, w, y ), m H =L( ml, mh, mw, my ) and H /m = L( l/m, h/m, w/m, y/m ) whenever m | gcd( H ). Theorem 2
Let H be a minimum distance diagram. Consider m ∈ N with m (cid:54) = 0 . Then,(a) m H is a minimum distance diagram.(b) If m | gcd( H ) , then H /m is a minimum distance diagram. Proof : Let us assume gcd( H ) = g , so the area of H is N with g | N and l = gl (cid:48) , h = gh (cid:48) , w = gw (cid:48) , y = gy (cid:48) with gcd( l (cid:48) , h (cid:48) , w (cid:48) , y (cid:48) ) = 1. The Smith normal form of the matrix M ( l, h, w, y ) is S = diag( g, N/g ) = U M V and the related 2–Cayley digraph is isomorphic toΓ = Cay( Z g ⊕ Z N/g , { a, b } ) with a = ( u , u ) and b = ( u , u ). Since H is a minimumdistance diagram, it fulfills Theorem 1, i.e. lh − wy = N , la = yb and hb = wa in Z g ⊕ Z N/g and ( l − y )( h − w ) ≥ m ∈ N with m (cid:54) = 0. Let us consider m H of area m N . The Smith normal form ofthe matrix mM is mS = U ( mM ) V . Let us consider the group G m = Z mg ⊕ Z m ( N/g ) . Nowwe have m ( la − yb ) = 0 and m ( hb − wa ) = 0 in G m and ( ml − my )( mh − mw ) ≥ m H fulfills Theorem 1 and it is a minimum distancediagram (related to Cay(G m , { a, b } )).Similar arguments can be used to prove that H /m is also a minimum distance diagram relatedto Cay( Z g/m ⊕ Z ( Ng ) /m ) (whenever m | g ). (cid:3) Now we define the procedures that give the kind of extensions and quotients we study in thiswork.
Definition 4 (Procedures E and Q ) Let H be a minimum distance diagram of area N ,with gcd( H ) = g ≥ , related to the digraph Γ = Cay( Z g ⊕ Z N/g , { a, b } ) . • Procedure E . We call the digraph m Γ , related to m H , the m - extension of Γ . • Procedure Q . For m | g , we call the digraph Γ /m , related to H /m , the m - quotient of Γ . By analogy, we also call the MDD m H the m -extension of H . We also call the MDD H /m the m -quotient of H whenever Procedure Q can be applied to H . Definition 4 is a correctdefinition by the following proposition. Proposition 1
Let us consider the digraph
Γ = Cay( Z g ⊕ Z N/g , { ( u , u ) , ( u , u ) } ) withrelated minimum distance diagram H of area N and gcd( H ) = g ≥ . Then,(a) For any m ∈ N , m (cid:54) = 0 , the m -expansion given by Procedure E , related to m H , is m Γ = Cay( Z mg ⊕ Z ( mN ) /g , { ( u , u ) , ( u , u ) } ) .(b) Let m ∈ N be a divisor of g . Then, the m -quotient given by Procedure Q , related to H /m , is Γ /m = Cay( Z g/m ⊕ Z N/ ( gm ) , { ( u , u ) , ( u , u ) } ) . Proof : The proof of these facts are a direct consequence of Theorem 2. (cid:3) , i, j ]] ∈ H = L( l, h, w, y ) by d([[ i, j ]]) = i + j .The value d([[ i, j ]]) represents the distance from the vertex 0 to the vertex ia + jb in the related2–Cayley digraph. We remark two important unit squares in H , p = [[ l − , h − y − q = [[ l − w − , h − H = max { d( p ) , d( q ) } . For instance, when consideringthe minimum distance diagram L(5 , , ,
2) of the left hand side of Figure 1, the unit square p corresponds to vertex 13 and q corresponds to 3. Lemma 1
Let H = L( l, h, w, y ) be a minimum distance diagram of area N and gcd( H ) = g > . Assume m | g with m ∈ N . Let us consider the unit squares p (cid:48) = [[ l/m − , h/m − y/m − and q (cid:48) = [[ l/m − w/m − , h/m − of the m -quotient H /m . Thus,(a) if d H = d( p ) , then d H /m = d( p (cid:48) ) ,(b) if d H = d( q ) , then d H /m = d( q (cid:48) ) . Proof : (a) We have d H = l + h − min { w, y } − l + h − w − p ). Then, d H /m = l/m + h/m − w/m − p (cid:48) ). The same argument proves item (b). (cid:3) Lemma 2
For a minimum distance diagram H and an m –extension m H , the equality d m H = m (d H + 2) − holds. Proof : This identity is a direct consequence of Lemma 1. (cid:3)
Notice that this lemma also states the identity d H /m = d H +2 m − Theorem 3
Quotients of optimal digraphs given by Procedure Q are also optimal digraphs. Proof : Let us assume that Γ is an optimal 2–Cayley digraph of order N related to theminimum distance diagram H . Let Γ (cid:48) = Γ /m be an m -quotient of Γ, generated by applyingProcedure Q . Thus, Γ (cid:48) has order N/m .Let us assume there is some 2–Cayley digraph of order N/m , ∆ (cid:48) , with related MDD L (cid:48) andd L (cid:48) < d H (cid:48) , where H (cid:48) = H /m . Let us consider the extension ∆ = m ∆ (cid:48) given by Procedure E .This extension has order N and diameter m (d L (cid:48) + 2) − < m (d H (cid:48) + 2) −
2. This fact leads tocontradiction because the diameter of Γ, m (d H (cid:48) + 2) −
2, is the smallest one over all 2–Cayleydigraphs of order N . (cid:3) This theorem ensures the optimality of a quotient of any optimal minimum distance diagram.Example 3 shows that this property is not true for extensions generated by Procedure E .Thus, there is a need to study when optimal extensions are obtained. Some properties oftight extensions are studied in the next section. In this section we are interested in studying optimal extensions obtained by Procedure E . Wefocus our attention on tight digraphs, i.e. tight extensions of tight digraphs. Proposition 2
Let us consider a tight minimum distance diagram H of area N and m ≥ .Then, m H is tight if and only if equality m (cid:100)√ N (cid:101) = (cid:100) m √ N (cid:101) holds. Proof : H is tight if and only if d H = (cid:100)√ N (cid:101) −
2. From Lemma 2 it follows that m H is tightif and only if equality m (cid:100)√ N (cid:101) = (cid:100) m √ N (cid:101) holds. (cid:3) { x } be defined by { x } = (cid:100) x (cid:101) − x . Lemma 3
Identity m (cid:100)√ N (cid:101) = (cid:100) m √ N (cid:101) holds for all m ≥ if and only if N = 3 t , for any t ≥ . Proof : Clearly identity m (cid:100)√ N (cid:101) = (cid:100) m √ N (cid:101) holds for all m ≥ N = 3 t .Assume now the identity holds for all m ≥
2. If √ N / ∈ N , then 0 < {√ N } <
1. Therefore,there is some large enough value m ∈ N with m {√ N } >
1. So, from identity m (cid:100)√ N (cid:101) = m √ N + m {√ N } , the inequality m (cid:100)√ N (cid:101) > (cid:100) m √ N (cid:101) holds. A contradiction. Thus,identity 3 N = x must be satisfied for some x ∈ N and so 3 | x . Hence, we have x = 3 t forsome t ∈ N and N = 3 t . (cid:3) Lemma 3 suggests the existence of an infinite family of tight non-cyclic 2–Cayley digraphs thatare t –extensions of a digraph on three vertices. The following result confirms this suggestion. Theorem 4
Let us consider the tight digraph Γ = Cay( Z , { , } ) with related minimumdistance diagram H = L(2 , , , . Then, for all t ≥ (a) H t = t H is a tight minimum distance diagram of area N t = 3 t ,(b) H t is related to Γ t = Cay( Z t ⊕ Z t , { (1 , − , (0 , } ) ,(c) D(Γ t ) = D ( N t ) = lb( N t ) = 3 t − . Proof : By Theorem 2, H t = L(2 t, t, t, t ) of area N t = 3 t , is a minimum distance diagramof Γ t for all t ≥
1. By Lemma 3 and Proposition 2, H t is tight for all t ≥
1. Thus (a) holds.Statement (b) comes from the isomorphism of digraphs Γ ∼ = Cay( Z ⊕ Z , { (1 , − , (0 , } )and then, Γ t is a t –extension of Γ . Statement (c) follows directly from the tightness of H t given by Proposition 2. (cid:3) By Lemma 3, the number of tight extensions of a tight digraph on N vertices is always finitewhenever N (cid:54) = 3 t . Now we are interested in the number of these tight extensions. Theorem 5
Let H be a tight minimum distance diagram of area N (cid:54) = 3 t . Then, the exten-sion m H is tight if and only if ≤ m ≤ (cid:106) (cid:100)√ N (cid:101)−√ N (cid:107) . Proof : Let Q be the set of rational numbers. If N (cid:54) = 3 t , then (cid:100)√ N (cid:101) − √ N / ∈ Q . Thus,there is some n ∈ N with n < (cid:100)√ N (cid:101)−√ N < n + 1. Hence, taking m ∈ N such that0 < n +1 < (cid:100)√ N (cid:101) − √ N < n ≤ m , inequalities 0 < m (cid:100)√ N (cid:101) − m √ N < m (cid:100)√ N (cid:101) = (cid:100) m √ N (cid:101) holds for m ≤ n = (cid:106) (cid:100)√ N (cid:101)−√ N (cid:107) .Assume now that m ≥ n + 1. Let us see that m H is not tight. From1 m ≤ n + 1 < (cid:100)√ N (cid:101) − √ N < n it follows that 1 < m (cid:100)√ N (cid:101) − m √ N . Since (cid:100) m √ N (cid:101) < m (cid:100)√ N (cid:101) , the extension m H is nottight by Proposition 2. (cid:3) Definition 5
Given N (cid:54) = 3 t , we define the extension coefficient c( N ) = (cid:106) (cid:100)√ N (cid:101)−√ N (cid:107) .
8y Theorem 5, the number of tight extensions of a tight digraph only depends on its tight-ness and its order N . It is a surprising fact that this coefficient does not depend on thestructure of the related group. For instance, consider the non isomorphic tight digraphs Γ =Cay( Z , { , } ) ∼ = Cay( Z ⊕ Z , { (0 , , ( − , } ) and ∆ = Cay( Z ⊕ Z , { (0 , , (1 , } ).Since c(189) = 5, Γ and ∆ have four tight extensions m Γ = Cay( Z m ⊕ Z m , { (0 , , ( − , } )and m ∆ = Cay( Z m ⊕ Z m , { (0 , , (1 , } ) for m ∈ { , , , } , respectively.
50 100 150 200 250 3001020304050
Figure 2: Values of c( N ) for 4 ≤ N ≤
300 and N (cid:54) = 3 t Clearly we can choose N (cid:54) = 3 t with small value of (cid:100)√ N (cid:101) − √ N , i.e. with large extensioncoefficient c( N ). Figure 2 shows some values of c( N ). This figure appears to suggest thatlargest values of c( N ) may have order O ( √ N ). Proposition 3 confirms this numerical evidence. Lemma 4 ([5, Proposition 3.1])
Set N = (cid:83) ∞ t =0 J t with J t = [3 t + 1 , t + 1) ] . Considerthe union J t = I t, ∪ I t, ∪ I t, with I t, = [3 t + 1 , t + 2 t ] , I t, = [3 t + 2 t + 1 , t + 4 t + 1] and I t, = [3 t + 4 t + 2 , t + 1) ] . Then (cid:100)√ N (cid:101) = t + 1 if N ∈ I t, , t + 2 if N ∈ I t, , t + 3 if N ∈ I t, . Proposition 3
Set E t,i = max { c( N ) : N ∈ I t,i and N (cid:54) = 3 k } for i ∈ { , , } . Then E t, = 6 t + 1 , E t, = 6 t + 3 and E t, = 2 t + 1 . Proof : Let us see E t, = 6 t + 1. Take N ∈ I t, with N (cid:54) = 3 k for all k ∈ N . Then, byLemma 4, we have (cid:100)√ N (cid:101) = 3 t + 1. Thus, (cid:100)√ N (cid:101) − √ N ≥ t + 1 − √ t + 6 t = α ( t ). Usingthe Mean Value Theorem, we have α ( t ) = (cid:112) (3 t + 1) − (cid:112) t + 6 t = 12 √ ξ t , with 9 t + 6 t < ξ t < (3 t + 1) . Then, from inequalities 6 t + 1 < √ t + 6 t < √ ξ t < t + 2, it follows that1 (cid:100)√ N (cid:101) − √ N ≤ α ( t ) = 2 (cid:112) ξ t , for N ∈ I t, , N (cid:54) = 3 k . Therefore, c( N ) ≤ (cid:98) √ ξ t (cid:99) = 6 t + 1 = c(3 t + 2 t ) for N ∈ I t, and N (cid:54) = 3 k . So E t, = 6 t + 1.Similar arguments lead to E t, = 6 t + 3 = c(3 t + 4 t + 1) and E t, = 2 t + 1 = c(3( t + 1) − (cid:3) N , i.e. E t, , E t, and E t, are only attained by N t, = 3 t +2 t , N t, = 3 t +4 t +1 and N t, = 3 t +6 t +2,respectively. Digraphs attaining the maximum number of consecutive tight extensions E t, , E t, and E t, are given in the following result. Theorem 6
Set N t, = 3 t + 2 t , N t, = 3 t + 4 t + 1 and N t, = 3 t + 6 t + 2 for t ≥ . Then,the tight digraphs attaining tight E t, , E t, and E t, -extensions are, respectively,(a) Γ t, = Cay( Z N t, , { t, t + 1 } ) ,(b) Γ t, = Cay( Z N t, , { t, t + 1 } ) (c) and Γ t, = Cay( Z N t, , { t + 1 , t } ) . Proof : Let us consider N t, = 3 t + 2 t and the L–shape H t, = L(2 t + 1 , t, t, t ). Then, thearea of H t, is N t, , gcd( H t, ) = 1 and the Smith normal form S t, of M t, = (cid:18) t + 1 − t − t t (cid:19) and the related unimodular matrices are S t, = diag(1 , N t, ) = U t, M t, V t, = (cid:18) t t + 1 (cid:19) M t, (cid:18) − t (cid:19) . Taking the generators a t = t and b t = 2 t + 1 and the digraph Γ t, = Cay( Z N t, , { a t , b t } ), thediagram H t, is related to Γ t, by Theorem 1. From equality d H t, = 3 t − N t, ), it followsthat the digraph Γ t, is tight. By Proposition 3, the digraph Γ t, has c( N t, ) = E t, = 6 t + 1consecutive tight extensions m Γ t, = Cay( Z m ⊕ Z mN t, , { (1 , t ) , (2 , t +1) } ), for 1 ≤ m ≤ t +1.Take now N t, = 3 t + 4 t + 1, the L–shape H t, = L(2 t + 1 , t + 1 , t, t ) and the matrix M t, = (cid:18) t + 1 − t − t t + 1 (cid:19) with Smith normal form S t, = diag(1 , N t, ) = (cid:18) t t + 1 (cid:19) M t, (cid:18) − t −
20 1 (cid:19) . Similar arguments as in the previous case lead to the tightness of the related digraph Γ t, =Cay( Z N t, , { t, t + 1 } ), with c( N t, ) = 6 t + 3 consecutive tight extensions m Γ t, = Cay( Z m ⊕ Z mN t, , { (1 , t ) , (2 , t + 1) } ).Finally, for N t, = 3 t + 6 t + 2, taking the MDD H t, = L(2 t + 2 , t + 1 , t, t ) with d H t, =lb( N t, ) = 3 t + 1 and the Smith normal form decomposition S t, = diag(1 , N t, ) with uni-modular matrices U t, = (cid:18) t + 1 t (cid:19) and V t, = (cid:18) − t − (cid:19) , we get the tight di-graph Γ t, = Cay( Z N t, , { t + 1 , t } ) that has c( N t, ) = 2 t + 1 consecutive tight extensions m Γ t, = Cay( Z m ⊕ Z mN t, , { (2 , t + 1) , (1 , t ) } ). (cid:3) A tight upper bound for the order of 2–Cayley digraphs, with respect to the diameter k , isknown to be AC ,k = (cid:106) ( k +2) (cid:107) (see for instance Dougherty and Faber [4] and Miller and ˇSir´aˇn[10]). It is worth mentioning that Γ t, and Γ t, attain this bound for k ≡ k ≡ k ≡ N t, + 1 = 3( t + 1) which hasinfinite tight extensions (Theorem 4). 10 Diameter improvement techniques
Two techniques for obtaining 2–Cayley digraphs with good diameter are given in this section.They are based on Procedure- Q and Procedure- E . The first one, known as E -technique ( Exten-sion technique), gives new tight 2–Cayley digraphs. The second one, known as QE -technique( Quotient-Extension technique), gives a 2–Cayley digraph which improves, if possible, thediameter of a given double-loop network of non square-free order.
Corollary 1 ( E -technique) Let us assume that G t is a family of tight –Cayley digraphs oforder N t , for t ≥ t . If m t = c( N t ) ≥ , for t ≥ t , then mG t is a family of tight –Cayleydigraphs for t ≥ t and ≤ m ≤ m t . Proof : The proof is a direct consequence of Theorem 5. (cid:3)
The first example of applying this technique is included in the proof of Theorem 6. Extensionsappearing in that proof are all tight ones and they are summarized in the following result.
Theorem 7
Given t ≥ , the following families contain tight digraphs over non-cyclic groups(i) Cay( Z m ⊕ Z mN t, , { (1 , t ) , (2 , t + 1) } ) for ≤ m ≤ t + 1 ,(ii) Cay( Z m ⊕ Z mN t, , { (1 , t ) , (2 , t + 1) } ) for ≤ m ≤ t + 3 ,(iii) Cay( Z m ⊕ Z mN t, , { (2 , t + 1) , (1 , t ) } ) for ≤ m ≤ t + 1 . The second example is given by Table 2.Digraph Order DiameterCay( Z ⊕ Z t +2 , { (3 t, − t + 1) , ( − , } ) 2 (12 t + 1) 12 t Cay( Z ⊕ Z t +4 t +2 , { (1 , − t ) , (0 , } ) 2 (3 t + 2 t + 1) 6 t + 2Cay( Z ⊕ Z t +8 t +4 , { (1 , − t − , (0 , } ) 2 (3 t + 4 t + 2) 6 t + 4Table 2: New optimal 2–Cayley digraphs using Procedure E Theorem 8
Table 2 gives three families of tight digraphs for all t ≥ . Proof : Consider the following tight families of double-loop networks of table [5, Table 2]Digraph Order Diameter G ,t = Cay( Z t +1 , {− t + 1 , } ) 12 t + 1 6 t − G ,t = Cay( Z t +2 t +1 , {− t, } ) 3 t + 2 t + 1 3 tG ,t = Cay( Z t +4 t +2 , {− t − , } ) 3 t + 4 t + 2 3 t + 1 G ,t corresponds to the entry 1.1 of table [5, Table 2] for x = 2 t . They are related to theminimum distance diagrams H ,t = L(4 t, t, t − , t + 1), H ,t = L(2 t + 2 , t + 1 , t, t + 2) and H ,t = L(2 t + 1 , t + 2 , t, t + 2), respectively. Digraphs of Table 2 are 2-extensions of thesedouble-loop networks. By Theorem 5, these extensions are also tight if and only if the valueof the expansion coefficient of G i,t is at least 2, for each i ∈ { , , } and t ≥ (12 t + 1). The other cases can be provedby similar arguments and are not included here. Considering G ,t and the related matrix M ,t (4 t, t, t − , t + 1), given in [5, Table 2], we use here the technique of Smith normal11orm explained in page 4. The Smith normal form of M ,t is S ,t = diag(1 , t + 1) and itcan be factorized as S ,t = U ,t M ,t V ,t = (cid:18) t − − t + 1 2 (cid:19) M ,t (cid:18) t + t t + 2 t + 1 (cid:19) , where U ,t and V ,t are unimodular integral matrices. Thus, G ,t related to H ,t is isomorphicto Cay( Z ⊕ Z t +1 , { (3 t, − t + 1) , ( − , } ).Let us compute now the expansion coefficient c( N ,t ), where N ,t = 12 t + 1. From thetightness of G ,t , we know D( G ,t ) = lb( N ,t ) = 6 t −
1, thus (cid:6)(cid:112) N ,t (cid:7) = 6 t + 1. Usingsimilar arguments as in the proof of Proposition 3, we have (cid:6)(cid:112) N ,t (cid:7) − (cid:112) N ,t = t − √ ξ t ,where 36 t + 3 < ξ t < (6 t + 1) . Then, it follows that 1 < t t − < √ ξ t t − < t +16 t − <
2. Therefore,c( N ,t ) = 2 for all t ≥
1. So, by Theorem 5, the 2-extension 2 G ,t = Cay( Z t +1 , {− t +1 , } )is also tight. (cid:3) Optimal double-loop networks in the bibliography are candidates to be improved whenevertheir orders are not square-free. Their optimality is restricted to cyclic groups. Thus, manyresults can be improved by considering 2–Cayley digraphs of the same order over non-cyclicgroups. The technique used in this case is a combination of quotients and extensions of2–Cayley digraphs. This task is detailed in the following result.
Corollary 2 ( QE -technique) Let Γ be a k -tight –Cayley digraph of order N , non square-free. Assume N = N (cid:48) m , m ≥ . If there is some minimum distance diagram H of area N (cid:48) such that d H < lb( N )+ k +2 m − , then the m -extension m H gives a –Cayley digraph ∆ with D(∆) < D(Γ) . Proof : Let us assume L is a minimum distance diagram related to Γ. Then, we have m | gcd( L ) and we can consider the m -quotient L /m . By Lemma 2, the diameter of L /m is d L /m = d L +2 m − D( G )+2 m − lb( N )+ k +2 m −
2. Thus, the existence of a minimumdistance diagram of area N (cid:48) and diameter d H < d L /m is equivalent to the existence of a2–Cayley digraph (related to the m -expansion m H ) ∆ with diameter D(∆) = d m H < D(Γ)(by Lemma 2 again). Finally, we haved H < d L /m ⇔ d H < lb( N ) + 2 m − . (cid:3) In order to remark how this technique can be used to improve some known results, twoexamples of different type are included here. The first one is of numerical nature and thesecond one is not.
Numerical example . Table 3 summarizes some numerical improvements as an example ofapplying the QE -technique. The entries of this table are ‘R’ the bibliographic cite, ‘ G or N ’original optimal double-loop or its order, ‘ T ’ tightness of G , ‘ G (cid:48) ’ new proposed 2–Cayley di-graph, ‘ T (cid:48) ’ tightness of G (cid:48) and ‘ m ’ that stands for the m -quotients and m -expansions requiredby Corollary 2.From the algorithmic point of view, the quotient reduces the time cost of searching theminimum distance diagram H mentioned in Corollary 2. The subsequent expansion is doneat constant time cost. 12 G or N T G (cid:48) T (cid:48) m [12, Remarks] Cay( Z , { , } ) 2 Cay( Z ⊕ Z , { (0 , , (1 , } ) 1 2” 3252 2 Cay( Z ⊕ Z , { (1 , , (1 , } ) 1 2” 3932 2 Cay( Z ⊕ Z , { (0 , , (1 , } ) 1 2” 4096 2 Cay( Z ⊕ Z , { (1 , , (0 , } ) 1 2” 4400 2 Cay( Z ⊕ Z , { (1 , , (2 , } ) 0 5” 4540 2 Cay( Z ⊕ Z , { (1 , , (1 , } ) 1 2” 4692 2 Cay( Z ⊕ Z , { (0 , , (1 , } ) 1 2” 5512 2 Cay( Z ⊕ Z , { (0 , , (1 , } ) 1 2” 3316 3 Cay( Z ⊕ Z , { (0 , , (1 , } ) 0 2” 21104 3 Cay( Z ⊕ Z , { (1 , , (3 , } ) 0 4” 23192 3 Cay( Z ⊕ Z , { (1 , , (0 , } ) 2 2[3, Lemma 4] Cay( Z , { , } ) 4 Cay( Z ⊕ Z , { (0 , , (1 , } ) 1 2” Cay( Z , { , } ) 4 Cay( Z ⊕ Z , { (1 , , (1 , } ) 3 2[3, Algorithm 2] 6505839 5 Cay( Z ⊕ Z , { (3 , , (7 , } ) 1 9” 8351836 5 Cay( Z ⊕ Z , { (1 , , (1 , } ) 0 2” 8568124 5 Cay( Z ⊕ Z , { (1 , , (1 , } ) 2 2” 8600936 5 Cay( Z ⊕ Z , { (0 , , (1 , } ) 2 2 Table 3: Some numerical improvements using the QE -technique R G T G (cid:48) T (cid:48) m [9, Table 1] Cay( Z t +2 t − , { a t , b t } ) 1 Cay( Z ⊕ Z e +32 e +40 , { a (cid:48) t , b (cid:48) t } ) 0 2 a t = 1 , b t = 3 t − e = 2 λ + 1, λ ≥ t = 2 e + 5, e ≥ a (cid:48) t = (1 , − λ − λ − , b (cid:48) t = (0 , Z ⊕ Z λ +32 λ +20 , { a (cid:48) t , b (cid:48) t } ) 0 2even e = 2 λ , λ ≥ a (cid:48) t = (1 , − λ − b (cid:48) t = ( − , λ + 2)[9, Table 2] Cay( Z t +4 t , { , e } ) 1 Cay( Z ⊕ Z e +4 e , { (1 , e + 1) , (0 , } ) 0 2even t = 2 e , e ≥ Z t +6 t +3 , { , t + 5 } ) 1 Γ t +1 of Theorem 4 for t ≥ Table 4: Some symbolical improvements using the QE -technique Symbolic example . QE -technique can also be applied as non-numerical task. An exampleof this feature is showed in Table 4. We check here the first two entries of Table 4, theother entries of the table can be checked by similar arguments. Replacing t = 2 e + 5 to N t = 3 t + 3 t −
5, we get N e = 2 (3 e + 16 e + 20). We have lb( N t ) = 3 t − e + 14 andD(Cay( Z t +2 t − , { a t , b t } )) = 3 t . Taking N (cid:48) e = 3 e + 16 e + 40 for odd e = 2 λ + 1, there is anMDD of area N (cid:48) e , H e = L(2 e +6 , e +4 , e +2 , e +2), that is tight d H e = lb( N (cid:48) e ) = 3 e +6 (noticethe identity 3 N (cid:48) e = (3 e + 8) − e = 2 λ + 1 and applying the Smith normalform procedure on H e , we obtain the generator set { (1 , − λ − λ − , (0 , } of the group Z ⊕ Z e +16 e +20 . Thus, after one 2-expansion, we get the 2–Cayley digraph G (cid:48) with diameterd H e = 2(3 e +6+2) − e +14. So, G (cid:48) is a tight digraph. When taking even e = 2 λ , there isa tight 2–Cayley digraph of order N (cid:48) e , F (cid:48) = Cay( Z ⊕ Z λ +16 λ +10 , { (1 , − λ − , ( − , λ + 2) } ).This digraph is tight and it is generated from the MDD L = L(4 λ +6 , λ +4 , λ +2 , λ +2), withgcd( L ) = 2. This diagram L comes from the diagram H e for e = 2 λ . After one 2-expansion,we obtain the second tight improvement of Table 4.13 Conclusion and future work
Minimum distance diagrams have been proved to be useful to define two digraph generationprocedures by quotient and extension. By Theorem 3, quotients generated by Procedure Q ofoptimal 2–Cayley digraphs on N (non square-free) vertices are also optimal diameter digraphs.Extensions generated by Procedure E of optimal digraphs are not always optimal.A characterization of tight Procedure E -like extensions have been found. Tight digraphswith infinite tight extensions have been also characterized and they have been proved to beextensions of the same digraph Γ = Cay( Z , { , } ) (Theorem 4). Those tight digraphs thatare not Procedure E -like extensions of Γ , with order N (cid:54) = 3 t , have been proved to havea finite number of consecutive tight extensions. This number has been called the extensioncoefficient and its exact expression has been found in Theorem 5. Theorem 6 shows that thereare infinite tight digraphs with extensions coefficient as large as wanted.These two procedures give two techniques that allow the generation of 2–Cayley digraphswhich can improve the diameter of many proposals in the literature. Several examples havebeen included.Now some future whishes. A quotient procedure with good metrical properties is neededwhen N is square-free. In this case, a study of the number of optimal quotients is worthstudying. A characterization of non-tight optimal extensions is also needed. References [1] F. Aguil´o and C. Mariju´an, Classification of numerical 3-semigroups by means of L-shapes
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