In this paper we characterize symmetric L-graphs, which are either Kronecker products of two cycles or Gaussian graphs.
Vertex symmetric networks have the property that the communication load is uniformly distributed on all the vertices so that
there is no point of congestion. A stronger notion of symmetry, edge symmetry, requires that every edge in the graph looks the
same. Such property ensures that the communication load is uniformly distributed over all the communication links, so that
there is no congestion at any link.Peer Reviewe

Beivide Palacio, Julio Ramón

Camarero, Cristóbal

Martínez, Carmen

In this paper we characterize symmetric L-graphs, which are either Kronecker products of two cycles or Gaussian graphs.

Vertex symmetric networks have the property that the communication load is uniformly distributed on all the vertices so that

there is no point of congestion. A stronger notion of symmetry, edge symmetry, requires that every edge in the graph looks the

same. Such property ensures that the communication load is uniformly distributed over all the communication links, so that

there is no congestion at any link.Peer Reviewe

UPCommons

Symmetric L-graphs

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Symmetric L-graphsCristo´bal Camarero, Carmen Mart´ınez and Ramo´n BeivideUniversidad de CantabriaSantanderAbstractIn this paper we characterize symmetric L-graphs, which areeither Kronecker products of two cycles or Gaussian graphs.Vertex symmetric networks have the property that the commu-nication load is uniformly distributed on all the vertices so thatthere is no point of congestion. A stronger notion of symmetry,edge symmetry, requires that every edge in the graph looks thesame. Such property ensures that the communication load isuniformly distributed over all the communication links, so thatthere is no congestion at any link.1 IntroductionMany interconnection networks have been based on vertex-transitive (orvertex-symmetric) graphs. This is the case of current parallel computersfrom Cray, HP and IBM, among others, that are built around torus net-works. Tori have displaced meshes that do not use wraparound edges whichsimpliﬁes planar design at the price of losing vertex-transitivity. Less workhas been devoted to edge-transitive (or edge-symmetric) networks. Squaretori would be the network of choice as it is symmetric (vertex and edgesymmetric). However, for practical reasons such as packaging, modularity,cost and scalability, the number of nodes per dimension might be diﬀerent.These topologies, are denoted as mixed-radix networks in [3]. Mixed-radixtori have the drawback of being non-edge-transitive which leads to an imbal-anced utilization of network links and buﬀers. For many traﬃc patterns,the load on the longer dimension is higher than on the shorter one and,hence, links in longer dimension become network bottlenecks, [1].169Symmetric L-graphs C. Camarero et al.The present work is devoted to ﬁnd and characterize the symmetricmembers of the two-dimensional family of undirected multidimensional cir-culant graphs, deﬁned in [5]. These graphs can be informally denoted asL-graphs, since their set of vertices is a subset of a two-dimensional lat-tice that can be represented as an L-shaped mesh with wrap-around edges.These two-dimensional networks can be considered as a generalization oftorus graphs as they have a planar representation when laying them out ona torus surface. L-shaped graphs have been widely considered, for examplein [2] and [6], with applications to interconnection networks.We recall now some deﬁnitions and results appeared in [5]. Let M ∈Mn×n(Z) be a n×n matrix of integers. Given two vectors a, b ∈ Zn we saythat a is congruent to b modulo M , which we denote as a ≡ b (mod M),if a − b ∈ MZn, where MZn stands for the additive group of column n-vectors with integral coordinates. We will also denote as Zn/MZn, thegroup of integral vectors modulo M , which has |det(M)| elements when Mis nonsingular.Deﬁnition 1 Let M ∈ M2×2(Z) be a non-singular matrix. Let A ={±a1,±a2} be a subset of vectors of Z2 such that {a1, a2} are linearlyindependent. The L-graph generated by M and adjacency A, L(M ;A), isdeﬁned as the graph whose vertex set is formed by the elements of Z2/MZ2and every vertex u is adjacent to u+ A (mod M).In particular, we are interested in those cases whose generating setof jumps is A = {e1, e2}, where e1 = (1, 0)t and e2 = (0, 1)t form theorthonormal two-dimensional basis. In general we will assume A = {e1, e2}and therefore we will just denote L(M ;A) = L(M).As stated above, in this paper we study the symmetry of L-graphs. Wewill proof that the only symmetric L-graphs are either Kronecker productsof two cycles or Gaussian graphs. Perfect codes over L-graphs were char-acterized in [7]. It was shown that L-graphs include Gaussian graphs [8],torus, twisted torus, Kronecker products of two cycles, etc. It is straightfor-ward that any Gaussian graph Ga+bi is isomorphic to L((a −bb a)). Kro-necker products of cycles have been proposed for interconnection networksin [11], [10], and codes over them were characterized in [12].Deﬁnition 2 Given a graph G = (V,E), Aut(G) denotes its automor-phisms group. G is said to be vertex-transitive if, for any pair of vertices170Symmetric L-graphs C. Camarero et al.v1, v2 ∈ V there exists f ∈ Aut(G) such that f(v1) = v2. Similarly, G issaid to be edge-transitive if for any pair of edges e1 = (v1, v2), e2 ∈ E thereexists f ∈ Aut(G) such that f(e1) = (f(v1), f(v2)) = e2. Then, if G is bothvertex and edge transitive, then it is called symmetric.It is easy to see that all L-graphs are vertex-transitive. Therefore, inorder to characterize symmetric L-graphs we will study those which areedge-transitive. With this aim, in Section 2 we will consider some propertiesof isomorphisms between L-graphs. In Section 3, we determine those L-graphs which are symmetric. In Subsection 3.1 we will focus on linearautomorphisms (group automorphisms of Z2/MZ2) of L-graphs which acttransitively on the set of edges in order to characterize those which are edge-transitive (and therefore symmetric) by means of their generator matrix.Finally, in Subsection 3.2 we analyze some marginal cases of symmetricL-graphs with non-linear automorphisms.2 Linear Automorphisms of L-graphsTwo multidimensional circulants are A´dam isomorphic if there exists anisomorphism between their groups of vertices such that sends the set ofjumps of one into the other one. It is clear that any A´dam isomorphicmultidimensional circulants are isomorphic, but the opposite it is not true.In [4] it was proved that:Theorem 3 [4] Any two ﬁnite isomorphic connected undirected Cayleymultigraphs of degree 4 coming from abelian groups are A´dam isomorphic,unless they are obtained with the groups and families Z4n, (1,−1, 2n +1, 2n − 1) and Z2n × Z2, ((1, 0), (−1, 0), (1, 1), (−1, 1)).In this Section we obtain a similar result in a diﬀerent form. We addressthe method here since it can be extended to higher dimensions, which isconsidered as future work.Deﬁnition 4 We deﬁne the neighborhood of a vertex v in the graph G =(V,E) as N(v) = {w : (v,w) ∈ E}. Then, the common neighborhood of alist of vertices denoted as v1, . . . , vn as N(v1, . . . , vn) =⋂ni=1 N(vi).171Symmetric L-graphs C. Camarero et al.Theorem 5 The neighborhood is preserved in graph isomorphisms. Thatis, if f is a graph isomorphism, thenN(f(v1), . . . , f(vn)) = {f(w) : w ∈ N(v1, . . . , vn)}.Proof: Let f be a graph isomorphism from G = (V,E) into G′ = (V ′, E′).We have that f(w) ∈ N(f(v1), . . . , f(vn)) if only if ∀i, f(w) ∈ N(f(vi)),that is ∀i, (f(w), f(vi)) ∈ E′. Since f is an isomorphism we have that thisis equivalent to ∀i, (w, vi) ∈ E so w ∈ N(v1, . . . , vn). Next, we analyze which multidimensional circulant graphs isomorphismsare linear mappings. We need the following:Deﬁnition 6 We say that a, b, c, d ∈ A form a 4-cycle in G(M ;A) if 0 =a + b + c + d1. Then, we say that G(M ;A) has not nontrivial 4-cycles ifa, b, c, d ∈ A such that 0 = a+b+c+d implies a = −b or a = −c or a = −d.Theorem 7 If G(M ;A) is a multidimensional circulant graph without non-trivial 4-cycles, then for all a, b ∈ A with a = bN(a, b) = {0, a + b}.Proof: If v ∈ N(a, b) then ∃x, y ∈ A such that v = a + x = b + y. Sincewe have a− b+x− y = 0 and G(M ;A) has not nontrivial 4-cycles, it mustbe fulﬁlled one of the following expressions:• a = b contradicting the hypothesis,• a = −x and thus v = a− a = 0,• a = y and thus v = b+ y = a + b. Lemma 8 Let G(M ;A) and G(M ′;A′) be two isomorphic multidimen-sional circulant graphs without nontrivial 4-cycles. Then any isomorphismf between G(M ;A) and G(M ′;A′) with f(0) = 0 is such that f(a + b) =f(a) + (b) for any a, b ∈ A with a = b.1each of {(v, v + a, v + a + b, v + a + b + c, v + a + b + c + d) : v ∈ V } is a cycle oflength 4172Symmetric L-graphs C. Camarero et al.Proof: Let a, b ∈ A with a = b. From the previous theorem we get thatN(a, b) = {0, a + b}, hence N(f(a), f(b)) = {f(0), f(a + b)} = {0, f(a) +f(b)}. As f(0) = 0 we have that f(a+ b) = f(a) + f(b). Lemma 9 In multidimensional circulant graphs translations are automor-phisms.Corollary 10 If f is an automorphism of G(M ;A), then there exists an-other automorphism f ′ with f ′(0) = 0.Lemma 11 Let f ∈ Aut(G(M ;A)) such that f(0) = 0 and G(M ;A) hasnot nontrivial 4-cycles. Then, we have that∀t ∈ G(M ;A),∀a, b ∈ A, f(t+ a+ b) = f(t + a) + f(t + b)− f(t).Proof: Let t ∈ G(M ;A) and a, b ∈ A. We can deﬁne ft(v) = f(t+v)−f(t),which is an automorphism with ft(0) = f(t) − f(t) = 0. Now if a = b,by Lemma 8 ∀t ∈ G(M ;A), ft(a + b) = ft(a) + ft(b), which implies∀t ∈ G(M ;A), f(t+ a+ b)− f(t) = f(t + a)− f(t) + f(t + b)− f(t).If a = b and a = −a then we have f(t+a−a)+f(t) = f(t+a)+f(t−a)and taking t = t′ + a, f(t′ + a) + f(t′ + a) = f(t′ + a + a) + f(t′).Finally if a = b = −a then let c = ft(a) and suppose c = −c. We havef−1t (c) = a, and because of Lemma 8, 0 = f−1t (c − c) = f−1t (c) + f−1t (−c).Hence f−1t (c) = −f−1t (−c) = a = −a = −f−1t (c), so f−1t (−c) = f−1t (c)and c = −c, which is a contradiction. Hence ft(a) = c implies c = −cand as consequence we obtain ft(a) = −ft(a). From this we get thatf(t+a)−f(t) = −f(t+a)+f(t), and taking t = t′+a that f(t′+a+a) =−f(t′ + 2a) + f(t′ + a) + f(t′ + a) and by 2a = 0 that f(t′ + a + a) =f(t′ + a) + f(t′ + a)− f(t′). Theorem 12 If the connected multidimensional circulant graph G(M ;A)has not nontrivial 4-cycles then any graph automorphism with f(0) = 0 isa group automorphism of Zn/MZn.Proof: We need to prove ∀ni ∈ N, ai ∈ A, f(∑i niai) =∑i nif(ai), forwhich is enough to see that for any v ∈ G(M ;A), a ∈ A, f(a + v) =173Symmetric L-graphs C. Camarero et al.f(a) + f(v). We proceed by induction in the number of summands of vin terms of elements of A. We have already proved that for 1 summand itholds. Assuming that it holds for N summands: let v be expressed withN + 1 summands. There exist b ∈ A, w ∈ G(M ;A) such that v = b + wwith w expressed with only N summands. Now, because of Lemma 11,f(a+ v) = f(w + a+ b) = f(w + a) + f(w + b)− f(w). And applying theinduction hypothesis we have that f(a+v) = f(w)+f(a)+f(w+b)−f(w) =f(a) + f(v). 3 Edge-transitivity of L-graphsIn this Section we determine those L-graphs which are edge-transitive.First, in Subsection 3.1 we consider the case where L(M) is an L-graphwithout nontrivial 4-cycles, that is, we can consider that any automor-phism of L(M) is a linear mapping. Later, in Subsection 3.2 we analyzethe special cases in which such cycles exist.3.1 Edge-transitive of L-graphs by Linear AutomorphismsIn this Subsection we will consider those L-graphs such that any automor-phism is a linear mapping. Some of the following results will be presentednot only for L-graphs but for any multidimensional circulant.Deﬁnition 13 A signed permutation is a composition of a permutationwith sign changing function.Deﬁnition 14 A signed permutation matrix is a matrix with entries in{−1, 0, 1} which has exactly one ±1 in each row and column.Note that in Zn×n the signed permutation matrices are exactly theunitary matrices, this is, the matrices U such UU t = I. They are relatedwith permutations in the way that for each signed permutation σ we canﬁnd a matrix Pσ such that⎛⎜⎝vσ(1)...vσ(n)⎞⎟⎠ = Pσ⎛⎜⎝v1...vn⎞⎟⎠174Symmetric L-graphs C. Camarero et al.Theorem 15 Let G(M) be a multidimensional circulant. Then, for eachlinear automorphism f there exists a signed permutation matrix P suchthat ∀α, f(α) = Pα.Proof: We deﬁne P as:Pi,j =⎧⎨⎩1 f(ej) = ei−1 f(ej) = −ei0 otherwisehaving f(ei) = Pei. Let α =∑niei.f(α) =∑inif(ei) =∑iniPei = P∑iniei = Pα. Theorem 16 Let G(M) be a multidimensional circulant. Then f(α) = Pαis a linear automorphism in G(M) if only if there exists a matrix Q suchthat PM = MQ.Proof: We prove ﬁrst the left to right implication. As f must be well-deﬁned, for all i, 0 = P0 ≡M PMei. And then exists qi such that PMei =Mqi, gathering all i’s togetherPM = [PMe1, . . . , PMen] = M [q1, . . . , qn] = MQNow, we prove the right to left implication. We begin proving f is well-deﬁned. Let a ≡M b, there is a γ such that a − b = Mγ, so Pa − Pb =PMγ = MQγ = Mγ′, getting Pa ≡M Pb. Now we prove injectivity.Let Pa − Pb = Mγ then a − b = P−1Mγ = MQ−1γ = Mγ′. And as|det(M)| = |det(PM)| the sizes are equal and so we have bijection. Andﬁnally we prove edge preservation. If a and b are connected in G(M) thenwe have ei and γ such a−b = ±ei+Mγ, and so Pa−Pb = ±Pei+PMγ =±ej + Mγ′ thus Pa is connected to Pb in L(M). To know if a multidimensional circulant G(M) without nontrivial 4-cycles is edge-transitive we need to look to the multiplicative group of thesigned permutation matrices P such PM = MQ. It is clear that if a matrixrepresenting a cycle of length n (even if it changes signs) is in the groupthen by composing it with itself, we can map e1 to every ei making the175Symmetric L-graphs C. Camarero et al.(a) Rotation (b) ReﬂexionFigure 1: Linear Automorphisms of L-graphsgraph edge-transitive. However, there exist groups which have not suchcycles but get the mapping, like the fourth alternating group.In dimension 2 this is simply to see that if for M one of(0 11 0),(0 −11 0)is in its group.Theorem 17 L(M) is edge-transitive if and only if M is right equivalentto one of the following matrices:•(a bb a), for P = Q =(0 11 0).•(a −bb a), for P = Q =(0 −11 0).•(a −ba b), for P =(0 −11 0), Q =(1 00 −1).Proof: The characteristic polynomial of(0 11 0)is λ2 − 1, and the one of(0 −11 0)is λ2 + 1. By Theorem III.2 in [9] it must be the characteristicpolynomial of both P and Q in PM = MQ. Therefore, we have two cases:176Symmetric L-graphs C. Camarero et al.• P =(0 11 0), λ2 − 1 = (λ + 1)(λ − 1), being reducible over Q, Qmust be similar to a matrix Q′ =(1 p0 −1), which is similar to either(1 00 −1)or(1 10 −1)S˜(0 11 0)• P =(0 −11 0), λ2+1 which is irreducible over Q and the class numberof Z[i] is 1, so Q must be similar to P (Theorem III.14, in [9]). It was proved in [7] that the Kronecker product of two cycles is alwaysan L-graph, as next Theorem shows.Theorem 18 [7] Let a, b ∈ Z. Then, the Kronecker product of the cyclesof lengths a and b, denoted as Ca ×Cb, is isomorphic to:• L((a+b2a−b2a−b2a+b2)) if a and b are odd integers such that a ≥ b.• 2 disjoint copies of L((a2−b2a2b2)) if a and b are both even integerssuch that a ≥ b.• L((a2 −ba2 b)), if a is an even integer and b is an odd integer.Therefore, L(M) in the ﬁrst and third cases of Theorem 17 is a Kro-necker product of two cycles and in the second one it is a Gaussian graph.3.2 Edge-transitive L-graphs by Nonlinear AutomorphismsIn this Subsection we focus on those L-graphs with nontrivial 4-cycles, thatis, L-graphs which can be isomorphic but not A´dam isomorphic. Clearly,if there is a nontrivial 4-cycle then there exist a, b ∈ A which fulﬁll:1. 4a ≡ 0 (mod M)2. 3a + b ≡ 0 (mod M)3. 2a + 2b ≡ 0 (mod M)177Symmetric L-graphs C. Camarero et al.If we consider ma+ nb = 0 (mod M) it means that there exists γ ∈ Z2such that κ =(mn)= Mγ. Now, let gcd(a1a2)= gcd(a1, a2), γ′ = γgcd γand κ′ = κgcd γ , having κ′ = Mγ′. As gcd γ′ = 1 we can build a unit matrixU with γ′ as one of its columns, and therefore M ′ = MU has κ′ as a column.We’ll begin with item (3). In this case we obtain the matrixM =(m 2n 2).If n = 2k we have that(m 22k 2)is right equivalent to(m− n −20 2). Onthe other hand, if n = 2k + 1 then(m 22k + 1 2)is right equivalent to(m− n− 1 −21 2). Both matrices generate the same graph and the iso-morphism is not linear in this case.If we analyze (1) and (2) in detail, we can obtain an equivalent toTheorem 12 for this case. In fact, we can ﬁnally prove that in these cases,all automorphisms are linear.Finally, there are a few marginal cases which correspond to those matri-ces M with both columns being nontrivial 4-cycles or with such a columnand 2ei in the other, that is, those which can be built by selecting twocolumns in the set:C ={(40),(31),(13),(04),(3−1),(1−3),(20),(02)}.A complete study of the following cases, shows as that most of thecombinations are edge-transitive. However, there are cases that lack of anon-linear automorphism, leading to non-edge-transitive graphs.Up to isomorphism, the L-graphs with 2 diﬀerent nontrivial solutionsto the 4-cycles are:• With nontrivial 4 cycles but without nonlinear automorphisms.(4 30 1),(4 10 3)• With nonlinear automorphism, which makes them edge-transitive,(4 00 2),(3 31 −1) (2 −10 3),(3 11 2)with an example in Figure 2.178Symmetric L-graphs C. Camarero et al.Figure 2: A nonlinear automorphism of L(M), where M =(2 −10 3)Figure 3: A nonlinear automorphism of the square torus of side 4179Symmetric L-graphs C. Camarero et al.• With nonlinear automorphism, but its linear automorphisms alreadymake them edge-transitive,(4 00 4),(3 11 3),(3 −11 3)with torus as example in Figure 3.AcknowledgementThis work has been supported by the Spanish Ministry of Education andScience under contract TIN2007-68023-C02-01.References[1] Camara, J., Moreto, M., Vallejo, E., Beivide, R., Miguel-Alonso, J.,Martinez, C., Navaridas, J. Twisted torus topologies for enhancedinterconnection networks. IEEE Transactions on Parallel and Dis-tributed Systems, 99 (PrePrints) (2010).[2] C. K. Wong, Don Coppersmith. A Combinatorial Problem Related toMultimodule Memory Organizations. Journal of the ACM, 21(3):392- 402, July 1974.[3] Dally, W., Towles, B. Principles and Practices of Interconnection Net-works. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA2003.[4] C. Delorme, O. Favaron and M. Mahe´o. Isomorphism of Cayley Multi-graphs of Degree 4 on Finite Abelian Groups. Europ. J. Combinatorics,13:59–61. 1992.[5] M. A. Fiol. On Congruence in Zn and the Dimension of a Multidimen-sional Circulant”. Discrete Mathematics, 141:123-134, 1995.[6] Fiol, M., Yebra, J. L., Alegre, I., and Valero, M. Discrete Optimiza-tion Problem in Local Networks and Data Alignment. IEEE Trans.Comput., 36(6):702-713, Jun. 1987.180Symmetric L-graphs C. Camarero et al.[7] C. Mart´ınez, C. Camarero, R. Beivide Perfect Graph Codes over TwoDimensional Lattices. Accepted for publication at 2010 IEEE Inter-nations Symposium on Information Theory.[8] C. Mart´ınez, R. Beivide, E. Staﬀord, M. Moreto´, E. M. Gabidulin.Modeling Toroidal Networks with the Gaussian Integers. IEEE Trans-actions on Computers, 57(8):1046–1056, August 2008.[9] M. Newman. Integral Matrices. Pure and Appl. Math. Series, Vol. 45.(Academic Press, New York, 1972.[10] Barak A. Pearlmutter. Doing the Twist: Diagonal Meshes Are Isomor-phic to Twisted Toroidal Meshes. IEEE Trans. Computers, 45(6):766-767, 1996.[11] K. Wendy Tang, Sanjay A. Padubidri. Diagonal and Toroidal MeshNetworks. IEEE Trans. Computers, 43(7):815-826, 1994.[12] J. Zerovnik. Perfect Codes in Direct Product of Cycles - a CompleteCharacterization. Advances in Applied Mathematics, 41:197-205, 2008.181

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