We present families of large undirected and directed Cayley graphs whose
construction is related to butterfly networks. One approach yields, for every
large $k$ and for values of $d$ taken from a large interval, the largest known
Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method
yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley
graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially
larger in $k$ than any previously constructed. In the directed case, these are
within a linear factor in $k$ of the Moore bound.Comment: 7 page

Bevan, David

English

arXiv

David Bevan

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Discrete Mathematics

Large butterfly Cayley graphs and digraphs

We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large k and for values of d taken from a large interval, the largest known Cayley graphs and digraphs of diameter k and degree d. Another method yields, for sufficiently large k and infinitely many values of d, Cayley graphs and digraphs of diameter k and degree d whose order is exponentially larger in k than any previously constructed. In the directed case, these are within a linear factor in k of the Moore bound

University of Strathclyde Institutional Repository

Bevan, David (2017) Large butterfly Cayley graphs and digraphs. Discrete Mathematics, 340 (10). pp. 2432-2436. ISSN 0012-365X , http://dx.doi.org/10.1016/j.disc.2017.05.012This version is available at https://strathprints.strath.ac.uk/60775/Strathprints is  designed  to  allow  users  to  access  the  research  output  of  the  University  of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may  not  engage  in  further  distribution  of  the  material  for  any  profitmaking  activities  or  any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge. Any correspondence concerning this service should be sent to the Strathprints administrator: strathprints@strath.ac.ukThe Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.arXiv:1507.08926v2  [math.CO]  16 May 2017Large butterfly Cayley graphs and digraphsDavid Bevan†AbstractWe present families of large undirected and directed Cayley graphs whose construction isrelated to butterfly networks. One approach yields, for every large k and for values of dtaken from a large interval, the largest known Cayley graphs and digraphs of diameter kand degree d. Another method yields, for sufficiently large k and infinitely many valuesof d, Cayley graphs and digraphs of diameter k and degree d whose order is exponentiallylarger in k than any previously constructed. In the directed case, these are within a linearfactor in k of the Moore bound.1 IntroductionThe goal of the degree–diameter problem is to determine the largest possible order of a graphor digraph, perhaps restricted to some special class, with given maximum (out)degree anddiameter. For an overview of progress on a wide variety of approaches to this problem, see thesurvey by Miller & Sˇira´nˇ [6].Our concern here is with large Cayley graphs and digraphs. Recall that, for a group G and aunit-free generating subset S of G, the Cayley digraph of G generated by S has vertex set G anda directed edge from g to gs for all g ∈ G and s ∈ S. If S is symmetric, i.e. S = S−1, then thecorresponding undirected simple graph is the Cayley graph of G generated by S. The Cayley(di)graph is thus regular of (out)degree |S| and vertex-transitive.We are interested in graphs and digraphs of degree d and diameter k, for arbitrary large k andvarying d. If a construction yields graphs of ordernd,k, we say that it has asymptotic order f(d, k)if, for fixed k,limd→∞nd,kf(d, k)= 1.No graph or digraph can be larger than the correspondingMoore bound. For undirected graphs,this bound is Md,k = 1 +dd−2((d − 1)k − 1)if d > 2. In the directed case, it is DMd,k =1d−1(dk+1 − 1)if d > 1. In both cases, the Moore bound has asymptotic order dk.†Department of Computer and Information Sciences, University of Strathclyde, Glasgow, Scotland.2010 Mathematics Subject Classification: 05C25, 05C20, 05C35.1Previously, for arbitrary degree and diameter, the largest known directed Cayley graphs wereobtained by Vetrı´k [7] and Abas & Vetrı´k [1], whose constructions have asymptotic order k(d2)kfor even k, and 2k(d2)kfor odd k. Our construction yields Cayley digraphs whose order isasymptotically kdk−1. For fixed diameter k > 8, these digraphs are larger than those in [7]and [1] for every value of d in a large interval. We also construct, for fixed k and infinitelymany values of d, Cayley digraphs whose asymptotic order is dke2k, a factor of 2k−1e2k2larger thanthose of Abas & Vetrı´k, and within a linear factor in k of the Moore bound.The undirected case is similar. Previously, the largest known Cayley graphs were obtained byMacbeth, Sˇiagiova´, Sˇira´nˇ & Vetrı´k [5], whose construction has asymptotic order k(d3)k. Ford − k 6≡ 3 (mod 4), we construct Cayley graphs whose order is asymptotically k(d2)k−1. Forsufficiently large diameter k, these graphs are larger than those in [5] for every suitable valueof d in a large interval. We also construct, for given k and infinitely many values of d, Cayleygraphs whose asymptotic order is 1e2k(d2)k, a factor of 1e2k2(32)klarger than those in [5].Our constructions are based on a two-parameter family of groups. For t > 2, let Zt = Z/tZ bethe additive group of integers modulo t, and for r > 2, let Z rt denote the product Zt × . . .× Zt,where Zt occurs r times, considered as an additive group of vectors. Let α be the automor-phism of Z rt , defined by α(v0, . . . , vr−1) = (vr−1, v0, . . . , vr−2), that cyclically shifts coordinatesrightwards by one, and consider the semidirect product G = Z rt ⋊ Zr, of order rtr, with thegroup operation given by (u, s)·(v, s′) = (u+αs(v), s+s′), for u, v ∈ Z rt and s, s′ ∈ Zr. We writeelements of G in the form (v0, . . . , vr−1; s), where each vi ∈ Zt and s ∈ Zr. Using this notation,the group operation is(u0, . . . ,ur−1; s)·(v0, . . . , vr−1; s′)= (u0 + vr−s, . . . , us−1 + vr−1, us + v0, . . . , ur−1 + vr−1−s; s+ s′),arithmetic in the subscripts being performed modulo r. The group G is used to create all ourCayley graphs and digraphs.The Cayley digraph generated by elements ofG of the form (a, 0, . . . , 0; 1), a ∈ Zt is isomorphicto the base-t order-r (wrapped) butterfly network, Bt(r), so called because it is composed of rtr−1edge-disjoint t-butterflies (copies of the complete bipartite graphKt,t); see [2, Figure 2]. Butterflynetworks are closely related to the de Bruijn graphs [3], the directed base-t order-r de Bruijngraph being a coset graph of Bt(r) [2, Theorem 4.4].Cayley graphs and digraphs ofGwere used previously byMacbeth, Sˇiagiova´, Sˇira´nˇ & Vetrı´k [5]and Vetrı´k [7] in the constructions mentioned above, though in neither case is the connectionto the butterfly networks made explicit. Each of our results is a consequence of choosing anappropriate set of generators for G. We make use of two distinct constructions.22 The first constructionWe present the directed case first, since it is slightly simpler.Theorem 1. For any k > 4 and d > k − 1, there exist Cayley digraphs that have diameter k,outdegree d, and order (k− 1)(d − k+ 3)k−1.Proof. Let r = k − 1 and t = d − k + 3, and let the underlying group of the Cayley digraph beG = Z rt ⋊ Zr. The order of G is rtr = (k − 1)(d − k+ 3)k−1.As generators for the Cayley digraph we use the t shift and add elements (a, 0, . . . , 0; 1), foreach a ∈ Zt, together with the remaining r − 2 nonzero cyclic shift elements (0, . . . , 0; s), for2 6 s 6 r − 1. Thus the digraph has outdegree t+ r− 2 = d.It also has diameter r + 1 = k. Every element is the product of r shift and add operations(establishing the vector) and possibly a single cyclic shift (to establish the final shift value if it isnonzero). On the other hand, if s 6= 0 then (1, . . . , 1; s) cannot be obtained as a product of fewerthan k generators.Clearly, the butterfly network Bt(r) is a subdigraph of the Cayley digraph of Theorem 1. Theadditional edges in our construction, corresponding to the cyclic shift elements, consist of trvertex-disjoint copies of the complete digraph on r vertices with a directed r-cycle removed.Vetrı´k [7] presents, for any k > 3 and d > 4, a family of Cayley digraphs of diameter k, degreed,and order k⌊d2⌋k. For odd diameters, Abas & Vetrı´k [1] improve this result by a factor of two,constructing Cayley digraphs of diameter at most k and degree d of order 2k⌊d2⌋k. Clearly, forlarge enough d, these digraphs are bigger than those of Theorem 1. However, for any givendiameter k > 8, it can be confirmed (using a computer algebra system, or otherwise) that thedigraphs of Theorem 1 are larger than those of Vetrı´k and Abas & Vetrı´k if2k+ 2 ln k < d < 2k−1(1− 1k)− k2.For specific values of the degree, we can domuch better. If we set d = k2−3k, then the digraphsof Theorem 1 have orders at least DMd,k/ek, within a linear factor of the Moore bound, andexceeding those of Abas & Vetrı´k by a factor of at least 2k−1/ek2, which exceeds 1 for k > 9.For the undirected case, we simply add elements to the generating set to make it symmetric.Theorem 2. For any k > 5 and d > k such that d − k 6≡ 3 (mod 4), there exist Cayley graphs thathave diameter k, degree d, and order (k− 1)(⌊d−k2⌋+ 2)k−1.Proof. Let r = k−1 and t =⌊d−k2⌋+2, and letG = Z rt ⋊Zr. As generators for the Cayley graphof G we use the t elements (a, 0, . . . , 0; 1), along with their inverses (0, . . . , 0,−a;−1), and theremaining r−3 nonzero elements (0, . . . , 0; s) for 2 6 s 6 r−2. In addition, if d−k ≡ 1 (mod 4),in which case t is even, then the involution (0, . . . , 0, t2 ; 0) is also included as a generator.3Thus the graph has degree 2t+r−3+(d−k mod 2) = d. As in the directed case, it has diameterr + 1 = k. Every element is the product of k− 1 shift and add operations and possibly a singlecyclic shift. On the other hand, if s /∈ {−1, 0, 1} then (1, . . . , 1; s) cannot be obtained as a productof fewer than k generators, and G has such an element since r > 4.Macbeth, Sˇiagiova´, Sˇira´nˇ & Vetrı´k [5] present, for any k > 3 and d > 5, a family of Cayleygraphs with diameter at most k, degree d, and order no greater than k(d+13)k.1 Their construc-tions also use the groupG, with a different generating set. For large enough d, these graphs arebigger than those of Theorem 2. However, for k > 27, the graphs of Theorem 2 are larger thanthose of Macbeth, Sˇiagiova´, Sˇira´nˇ & Vetrı´k for any d− k 6≡ 3 (mod 4) satisfying3k+ 6 ln k < d < 2(32)k(1− 1k)− k2.For specific values of the degree, we can do much better. If we set d = k2 − 2k, then the graphsof Theorem 2 have orders exceeding those in [5] by a factor of at least 2ek2(32)k, which exceeds 1for k > 14.3 The second constructionIn our second construction, we conceive of the vectors of length r as being partitioned into k−1long blocks, each of length ℓ, and a single short block, of lengthm.Again, the directed case is presented first, since it is simpler.Theorem 3. For any k, ℓ, t > 2 and positivem < ℓ, there exist Cayley digraphs that have diameter k,outdegree tℓ + (r − 1)tm − 1, and order rtr, where r = (k− 1)ℓ +m.Proof. As before, let G = Z rt ⋊ Zr, of order rtr. As generators for the Cayley digraph, we usethe tℓ long elements (a1, . . . ,aℓ, 0, . . . , 0; ℓ), ai ∈ Zt, together with the additional (r − 1)tm − 1nonzero short elements (a1, . . . ,am, 0, . . . , 0; s), ai ∈ Zt, s 6= ℓ. Thus the digraph has outdegreetℓ + (r − 1)tm − 1. Long elements shift by ℓ and modify a long block; short elements shiftarbitrarily and modify a short block.The digraph has diameter k. Every element is the product of a single short element (establishingm components of the vector and guaranteeing the final shift value) and k − 1 long elements(establishing the remaining (k − 1)ℓ = r − m components of the vector). On the other hand,(1, . . . , 1; 0) cannot be obtained as a product of fewer than k generators.The Cayley digraph of Theorem 3 contains both of the butterfly networks Btℓ(r) and Btm(r)as subdigraphs. Its edges can be partitioned into rtr−ℓ copies of the tℓ-butterfly, from the longelements, r(r − 2)tr−m copies of the tm-butterfly, from the short elements that have nonzeroshift, and a collection of directed cycles from the short elements with zero shift.1The graphs in [5] are slightly larger than those ofMacbeth, Sˇiagiova´ & Sˇira´nˇ [4], whose order is at most k(d+13)k−k.4Given k, ℓ and t, for judicious choice of m, these digraphs are larger than those of Abas &Vetrı´k [1]. For example, if we let t = 2, then for all k > 31 and sufficiently large ℓ, the order ofour digraphs is greater than that of those in [1] ifℓ − k− log2ℓ + 2 < m < ℓ − log2kℓ−2k(log2k+ 2).Ifm is chosen optimally, we can do much better than that.Corollary 4. For any k > 3, there are arbitrarily large values of d for which there exist Cayleydigraphs that have diameter k, outdegree d, and order at least 1k(kk+2(d + 1))k.Proof. We use the construction of Theorem 3. Let t = 2, and let ℓ be any sufficiently largepositive integer such that log2 k2ℓ 6 34ℓ. Let r =⌈kℓ− log2 k2ℓ⌉, and m = r − (k − 1)ℓ, sor = (k− 1)ℓ +m. Note that 0 < m < ℓ.The digraph has diameter k and order r2r, which (rounding r down) is at leastn0 =(kℓ− log2 k2ℓ)2kℓ−log2 k2ℓ =(1k−log2 k2ℓk2ℓ)2kℓ.Its degree is d = 2ℓ + (r − 1)2m − 1, which (substituting form and rounding r up) is less thand+ = 2ℓ +(kℓ− log2 k2ℓ)2kℓ−log2 k2ℓ+1−(k−1)ℓ − 1 =(1+2k−2 log2 k2ℓk2ℓ)2ℓ − 1.Let θ =log2 k2ℓkℓ. Note that the condition on ℓ implies that θ 6 34k 614 , since k > 3.Now,kn0(kk+ 2(d+ + 1))−k= (1− θ)(1+2θk+ 2− 2θ)k> (1− θ)(1+2kθk+ 2− 2θ),which is at least 1 if k > 2 and 0 6 θ 6 k−22k−2 . Since k > 3 and θ 614 , the result follows.These digraphs have asymptotic order exceeding dke2k, a factor of 2k−1e2k2larger than those of Abas& Vetrı´k, and within a linear factor in k of the Moore bound.It is worth briefly explaining the choice of values for t and r in the proof of Corollary 4. Supposewe fix t and r (and hence the order rtr), and also fix the diameter k. What is the optimal choicefor ℓ, that minimises the degree tℓ + (r − 1)tr−(k−1)ℓ − 1? Differentiating with respect to ℓ andequating to zero yields ℓ = 1k(r + logt(k− 1)(r − 1)). Solving for r then givesr =1ln tW(tkℓ−1 ln tk− 1)+ 1,where W is the Lambert W function, defined implicitly by W(z)eW(z) = z. Asymptotically,W(z) = ln z − ln ln z + o(1). Applying this approximation for W then yields r ≈ kℓ − logtk2ℓ.5Using this value for r results in a digraph whose order is asymptotically at least 1k(kk+t(d+1))k.Setting t = 2 makes this maximal.The results in the undirected case are similar. As before, we just add elements to the generatingset to make it symmetric.Theorem 5. For any k, ℓ, t > 2 and positive m < ℓ, there exist Cayley graphs that have diameter k,degree 2tℓ + (2r − 3)tm − r, and order rtr, where r = (k − 1)ℓ +m.Proof. Let G = Z rt ⋊ Zr. As generators for the Cayley graph of G with these parameters, weuse:• the tℓ long elements (a1, . . . ,aℓ, 0, . . . , 0; ℓ), ai ∈ Zt• their tℓ inverses (0, . . . , 0,a1, . . . ,aℓ;−ℓ)• the (r − 2)(tm − 1) short elements (a1, . . . ,am, 0, . . . , 0; s), ai ∈ Zt not all zero, s /∈ {0, ℓ}• their (r − 2)(tm − 1) inverses (0, . . . , 0,s︷ ︸︸ ︷a1, . . . ,am, 0, . . . , 0;−s)• the tm − 1 nonzero short elements (a1, . . . ,am, 0, . . . , 0; 0); this set is symmetric• the r − 3 short elements (0, . . . , 0; s), s /∈ {0,±ℓ}; this set is also symmetricThus the graph has degree 2tℓ + (2r − 3)tm − r. As in the directed case, it has order rtr anddiameter k.Given k, ℓ and t, for appropriate choice of m, these graphs are larger than those of Macbeth,Sˇiagiova´, Sˇira´nˇ & Vetrı´k [5]. For example, if we let t = 2, then for all k > 69 and sufficientlylarge ℓ, the order of our graphs is greater than that of those in [5] ifℓ + k− log23kℓ+ 1 < m < ℓ − log2kℓ−3k(log2k+ 2) − 1.Ifm is chosen optimally, we have the following.Corollary 6. For any k > 3, there are arbitrarily large values of d for which there exist Cayley graphsthat have diameter k, degree d, and order at least1k(k2k+4(d + k log2d2 − log2 log2 d − log2 8k2))k.Proof. We use the construction of Theorem 5. As in the proof of Corollary 4, let t = 2, and let ℓbe any sufficiently large positive integer such that log2 k2ℓ 6 34ℓ. Let r =⌈kℓ− log2 k2ℓ⌉, andm = r− (k− 1)ℓ, so r = (k− 1)ℓ +m.The graph has diameter k and order r2r, which is at leastn0 =(kℓ− log2 k2ℓ)2kℓ−log2 k2ℓ =(1k−log2 k2ℓk2ℓ)2kℓ.6Its degree is d = 2ℓ+1 + (2r − 3)2m − r, which (substituting for m and rounding r up in thesecond term) is less than2ℓ+1 +(2kℓ− 2 log2 k2ℓ− 1)2kℓ−log2 k2ℓ+1−(k−1)ℓ − r =(2+4k−1+ 4 log2 k2ℓk2ℓ)2ℓ − r.Thus, 12(d + r) is less than q =(1+ 2k−2 log2 k2ℓk2ℓ)2ℓ, and by the argument in the proof ofCorollary 4 (with q = d+ + 1), we know that kn0 >(kqk+2)k>(k2k+4(d + r))k.It remains to establish the appropriate lower bound for r.Now, kn0 < 2kℓ and q > d2 , so 2ℓ > kd2k+4 and thus ℓ > log2kd2k+4 = log2d2 − log2(1+ 2k).Since(1+ 2k)k< e2 < 23, we have log2(1+ 2k)< 3kand thus ℓ > log2d2 −3k.Now, r > kℓ− log2 k2ℓ, sor > k log2d2 − 3 − log2 k2 − log2(log2d2 −3k),which is greater than k log2d2 − log2 log2 d− log2 8k2, as required.These graphs have asymptotic order exceeding 1e2k(d2)k, a factor of 1e2k2(32)klarger than thoseof Macbeth, Sˇiagiova´, Sˇira´nˇ & Vetrı´k.AcknowledgementsThe author is very grateful to Grahame Erskine for pointing out errors in earlier drafts, and alsoto an assiduous referee whose detailed comments greatly improved the quality of this paper.S.D.G.References[1] Marcel Abas and Toma´sˇ Vetrı´k. Large Cayley digraphs and bipartite Cayley digraphs of odd diameters. DiscreteMath., 340(6):1162–1171, 2017.[2] Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg. Group action graphs and parallel architectures.SIAM J. Comput., 19(3):544–569, 1990.[3] N. G. de Bruijn. A combinatorial problem. Nederl. Akad. Wetensch., Proc., 49:758–764, 1946.[4] Heather Macbeth, Jana Sˇiagiova´, and Jozef Sˇira´nˇ. Cayley graphs of given degree and diameter for cyclic,Abelian, and metacyclic groups. Discrete Math., 312(1):94–99, 2012.[5] Heather Macbeth, Jana Sˇiagiova´, Jozef Sˇira´nˇ, and Toma´sˇ Vetrı´k. Large Cayley graphs and vertex-transitivenon-Cayley graphs of given degree and diameter. J. Graph Theory, 64(2):87–98, 2010.[6] MirkaMiller and Jozef Sˇira´nˇ. Moore graphs and beyond: A survey of the degree/diameter problem. Electron. J.Combin., 20(2): Dynamic survery 14v2, 92 pp, 2013.[7] Toma´sˇ Vetrı´k. Large Cayley digraphs of given degree and diameter. Discrete Math., 312(2):472–475, 2012.7

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Large butterfly Cayley graphs and digraphs

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