A communication network can be modelled as a graph or a directed graph, where each processing element is represented by a vertex and the connection between two processing elements is represented by an edge (or, in case of directed connections, by an arc). When designing a communication network, there are several criteria to be considered. For example, we can require an overall balance of the system. Given that all the processing elements have the same status, the flow of information and exchange of data between processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element, that is, if there is a balance (or regularity) in the network. This means that the in-degree and out-degree of each vertex in a directed graph (digraph) must be regular. In this paper, we present the existence of digraphs which are not diregular (regular out-degree, but not regular in-degree) with the number of vertices two less than the unobtainable upper bound for most values of out-degree and diameter, the so-called Moore bound

Miller, Mirka

Slamin, S

English

Jurnal ILMU DASAR

Jurnal ILMU DASAR Vol. 12 No. 1, Januari 2011 : 1-5 1On The Existence of Non-Diregular Digraphs of Order Two Less than the Moore Bound*Slamin1) & Mirka Miller2)1) Informatic System Department, Jember University Indonesia2) School of Electrical Engineering and Computer Science, The University of Newcastle, AustraliaDepartment of Mathematics, University of West Bohemia, Pilsen, Czech RepublicDepartment of Computer Science, King's College London, UKDepartment of Mathematics, ITB Bandung, IndonesiaABSTRACTA communication network can be modelled as a graph or a directed graph, where each processing element is represented by a vertex and the connection between two processing elements is represented by an edge (or, in case of directed connections, by an arc). When designing a communication network, there are several criteria to be considered. For example, we can require an overall balance of the system. Given that all the processing elements have the same status, the flow of information and exchange of data between processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element, that is, if there is a balance (or regularity) in the network. This means that the in-degree and out-degree of each vertex in a directed graph (digraph) must be regular. In this paper, we present the existence of digraphs which are not diregular (regular out-degree, but not regular in-degree) with the number of vertices two less than the unobtainable upper bound for most values of out-degree and diameter, the so-called Moore bound.Keywords : Non-diregular digraphs, Moore boundINTRODUCTIONLarge  communication  network  design  has become  a  growing  interest  due  to  recent advances  in  very  large  scale  integrated technology. In such networks, it is desirable to have  connections  which  achieve  the  most efficient and reliable communication in view of practical economic constraint.In communication network design, there are several  factors  which  should  be  considered. Two  of  the  factors  seem  to  appear  most frequently,  namely,  (a)  the  number  of connections  which  can  be  attached  to  a processing element is limited,  and (b) a short communication  route  between  any  two processing elements is required. We would like to end up with a large network subject to these constraints. Another factor that may be considered when designing  a  communication  network  is  fault tolerance.  The  fault  tolerance  of  a communication network is the capability of the network to continue working when a number of processing elements or connections have failed. The  larger  number  of  faulty  processing elements  or  connections  that  can be tolerated the better fault tolerance.     When designing a communication network, we  may  require  an  overall  balance  of  the system. Given that all the processing elements have the same status, the flow of information and  exchange  of  data  between  processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element, that is,  if  there  is  a  balance  (or  regularity)  in  the network.The  underlying  topology  of  a  network  is usually  studied  by  graph-theoretic  means:  a network  is  represented  by  a  graph  which consists of a set of vertices; some or all pairs of the vertices may be joined by edges (or, in case of  directed  connections,  by  arcs).  Each processor  is  represented  by a vertex and two particular  vertices  are  joined  by  an  edge (respectively,  arc)  if  there  is  a  two-way (respectively,  one-way)  connection  between the two corresponding processors.The  number  of  connections  incident  to vertex  A is  called  the  it  degree of  A;  if  the connections are one way only then we make a distinction  between  in-coming  and  out-going connections and speak of the in-degree and the out-degree of  A.  If every vertex has the same in-degree  (respectively,   out-degree)  then  the network is said to be  in-regular (respectively, out-regular).  If  the  network  (digraph)  is  in-regular  and  out-regular,  then  it  is  called diregular. The distance from vertex A to vertex B is the length of the shortest path from A to B measured by the number of edges (or arcs) that 2 On The Existence.....(Slamin & Mirka Miller)need to be traversed in order to reach B from A. The  maximum  distance  between  any  pair  of vertices is called the diameter of the graph.Among the many factors to be considered in a network design, there are two which seem to appear  most  frequently:  The  number  of connections that can be attached to a vertex is limited,  and  a  short  communication  route between  any  two  vertices  is  required. Naturally,  one  would  like  to  end  up  with  a large  network  subject  to  these  constraints.  In graph-theoretic  terms  this  corresponds  to  the following well-known fundamental problem.Degree/diameter problem Construct graphs of given maximum degree  d and  diameter  k with  the  largest  possible number of vertices.The directed version of the problem differs only in that `degree' is replaced by `out-degree' in  the  statement  of  the  problem.  We  treat separately  the  case  of  directed  networks because although the extremal problems arising there are superficially similar to the undirected case,  they  do  require  substantially  different techniques. A straightforward general  upper  bound on the  largest  possible  order  (i.e.  number  of vertices)  nd,k of a graph of maximum degree  d and diameter k is the Moore bound Md,k, named after  E.  F.  Moore  who  first  proposed  the problem: nd,k ≤  Md,k = 1 + d + d(d - 1) + … + d(d -1)k-1 for undirected graphs, andnd,k ≤  Md,k = 1 + d + d2 + … + dkfor directed graphs (digraphs).In general, these bounds cannot be attained. For the case of directed graphs, the bound can be obtained only when d = 1 or k = 1 (Bridges & Toueg   1980).  Since  there  is  no  digraph whose  order  attained  the  Moore  bound  for maximum out-degree  d ≥  2 and diameter  k ≥  2,  the  research  of  the  existence  of  large digraphs  focuses  on  digraphs  whose  order  is close to the Moore bound. The research on this area has been carried out by several world class researchers. Most of the  research  activities  focus  on the  two foll.owing mainstreams:proving the non-existence of digraphs of order `close'  to  the  Moore  bound,  for  example (Baskoro et al. 2005, Baskoro  et al. 1998, Hoffman & Singleton 1960,  Miller   & Siran 2001) constructing large digraphs, for example (Comellas & Fiol 1990, Fiol  et al. 1984, Miller & Slamin 2000).This  motivated  author  to  conduct  another research  activity  concerning  digraphs  with order  close  to  the  Moore  bound,  that  is,  the study of the regularity of such digraphs.  This paper  deals  mainly  with  the  issue  of diregularity of directed graphs (digraphs) with the  number  of  vertices  close  to  the unobtainable upper bound for  most  values  of out-degree d and diameter k.PREVIOUS RESULTSThe  study  of  diregularity  of  a  digraph  is conducted by proving in-regularity of digraph as well as its out-regularity. If such digraph is both  in-regular  and  out-regular,  then  the digraph is diregular.The  digraph  whose  order  is  equal  to  the Moore bound is called  Moore digraph. Moore digraph has  diameter  equal  to  k and the out-degree of each of its vertices equal to  d. It is easy  to  show  that  all  in-degrees  in  such  a Moore  digraph  must  be  equal  to  d.  These observations  follow  from  the  well  known classification  (Bridges  & Toueg  S  1980; Plesnik & Znam 1974) which says that Moore digraphs exist only in the trivial cases when d = 1 (directed cycles of length k + 1 for any k  ≥  1) or  k = 1 (complete digraphs of order  d + 1 for any d ≥  1). The research continued on the diregularity of digraph whose order is close to the Moore bound  (Md,k –  δ,  for  1  ≤  δ <  d).  The  out-regularity  of  digraphs  of  order  close  to  the Moore bound was observed in  (Baskoro et al. 1998).  We  state  the  observation  in  the following theorem. Theorem 1 [Baskoro,  Miller and Plesník]   The  digraph of  maximum out-degree d  ≥  2, diameter k ≥  2 and order Md,k – δ, for 1 ≤ δ < d, must be out-regular of out-degree d.Proof. Suppose  that  the  digraph  contains  a vertex u with out-degree d1 < d (i.e., d1 ≤  d-1), then considering the number of vertices in the out-bound spanning tree starting from vertex u, the order of the digraph: Jurnal ILMU DASAR Vol. 12 No. 1, Januari 2011 : 1-5 3dMnddddndddndddndddddnkdkkkkk−<+++−+++=+++−+≤++++=++++≤−−−−,11111111)...1()...1()...1)(1(1)...1(1...1Hence the out-degree of any vertex in a digraph of order Md,k – δ, for 1 ≤ δ < d, must be equal to d, that is, such a digraph must be out-regular.    In contrast, establishing the in-regularity of digraphs of order close to the Moore bound is not  as  easy  as  proving  the  out-regularity  of digraphs.  Figure  1  shows  the  fact  that  there exist a digraph of out-degree d = 2, diameter k = 2 and order  M2,2 – 2 as well as a digraph of out-degree d = 3, diameter k = 2 and order M3,2 – 3 in which not all vertices have the same in-degree.             G2  G3Figure 1. Non-diregular digraphs G2 and G3.Some  results  on  diregularity  of  digraph whose order is close to the Moore bound (n = Md,k – δ where 1 ≤ δ < d) have been established. Miller,  Gimbert,  Siran  and  Slamin (2000) proved that every digraph whose order is one less than Moore bound (n =  n =  Md,k – 1) is diregular (Miller et al. 2000).For the case of digraph whose order is two less than Moore bound (n =  Md,k – 2), Slamin and  Miller  have  proved  that  the  digraphs  of out-degree d = 2, diameter k > 2 and order n = Md,k –  2  are  diregular  (Slamin  & Miller  M 2000). For out-degree  d = 3, Slamin, Baskoro &  Miller (2001) have  investigated  some properties and structures of the diregularity of digraphs of order two less than Moore bound, however,  it  is  not  known  whether  the  such digraphs are diregular or not. In  this  paper,  we present  the  existence  of non-diregular  digraphs  of  out-degree  d ≥  2, diameter  k = 2 and order two less than Moore bound (n  = Md,2 –  2)  as  described  in  the following section.RESEARCH PROCEDUREThis  research  is  conducted  by  investigating  the known  results  on  the  construction  techniques  of large digraphs that have been established, especially vertex deletion scheme (Miller  & Slamin 2000). In detail, the procedure of the research is as follows.1. Observing the largest known digraphs given out-degree degree d ≥ 2, diameter k ≥ 2 and order n = dk + dk-12. Observing the  construction  techniques of  large digraphs using vertex deletion scheme.3. Applying  vertex  deletion  scheme  technique  to the largest known digraphs.4. Investigating  the  order  of  generated  digraphs obtained from vertex deletion scheme.Investigating the diregularity of generated digraphs obtained from vertex deletion scheme.MAIN RESULTSBefore  presenting  the  results,  we  present  the definition  of  Kautz  digraph  and  theorem  on construction  technique  of  digraph  so-called vertex deletion scheme. Definition  1.  Kautz  digraph,  denoted  by  K(d,k), is the diregular digraph of out-degree d ≥ 2, diameter k ≥ 2 and order n = dk + dk-1If the diameter of Kautz digraph is k = 2, then the order is  n =  d2 +  d  =  Md,2 – 1 for  d  ≥ 2. Kautz digraph can be obtained from complete digraph  by  using  line  digraph construction technique. This construction technique can be described as follows. Let  G = (V,A) and let  N be the set of all walks of length 2 in G. The line digraph  of  a  digraph  G,  L(G)  =  (A,N),  is  a digraph such that the set of vertices of L(G) is equal to the set of arcs of G and the set of arcs of L(G) is equal to the set of walks of length 2 in  G.  This means that  a vertex  uv of  L(G) is adjacent to a vertex wx if and only if v = w.The line digraph construction technique can be used to  construct  a  new  digraph  whose  order  is larger  than  the  original  digraph.  This construction  technique  provides  the diregularity  on  the  degree  of  the  original digraph  which  means  that  if  digraph  G is diregular of degree d then the line digraph L(G) is  also diregular  of degree  d.  Figure 2 shows the  complete  digraph  K5  and  Kautz  digraph Ka(4,2) of degree 4, diameter 2 and order  n = 42 + 4 = 20 which is obtained from  K5  using line digraph construction technique L(K5).  4 On The Existence.....(Slamin & Mirka Miller)Figure  2.  Complete  digraph  K5  and  its  line digraph Ka(4,2).Teorema 2. [Miller and Slamin]. If G is the digraph of out-degree d, diameter k and order n with property that out-neighbours of vertex u are  equal  to  the  out-neighbours  of  vertex  v, then  there  is  a  digraph  G’  of  out-degree   d, diameter at most k and order  n – 1. ðDigraph G’ is constructed from G by utilizing vertex deletion scheme. The procedure of this technique can be described as follows. Let  G be the digraph of out-degree d, diameter k and order  n.  Let  N  +(v)  denotes  the  set  of  out-neighbour vertices of vertex v.  If u and v are vertices in G with property that N +(u) = N +(v), then G’ can be obtained by deleting vertex u together  with  all  arcs  going  out  from  u  and reconnecting all incoming arcs  of vertex u to vertex v.Digraphs which are obtained from the line digraph iterations always contain some pairs of vertices  with  the  same  out-neighbourhoods. Since Kautz digraph Ka(d,k)  of out-degree d, diameter  k  and  order  dk +  dk-1 can  be constructed using the k-1 iterations line digraph of  complete  digraph  Kd+1,  we  can  apply Theorem 2 to obtain new digraphs with order less than Kautz digraphs. In  the  following  theorem  we  shall  prove that  the  new  digraphs  obtained  from  Kautz digraph using vertex deletion scheme are  not diregular. Teorema 3. There exist non-diregular digraphs of out-degree d ≥ 2, diameter k = 2, and order two less than Moore bound (n = Md,2 – 2).Proof. Let Ka(d,2) be the Kautz digraph of out-degree d ≥ 2 and diameter k = 2. By definition of Kautz digraph, the order of Ka(d,2) is n = d2 + d, that is one less than Moore bound or n = Md,2 – 1. Since  Ka(d,2) has d + 1 distinct sets containing  d  vertices  with  the  same  out-neighbourhoods,  so  we  can  choose  two vertices, say u and v with property that N +(u) = N  +(v). By Theorem 2, we can construct new digraph G’, by deleting vertex u together with all arcs going out from u and reconnecting all incoming arcs of vertex u to vertex v. The new digraph G’ has out-degree d, diameter at most k and order n – 1 = Md,2 – 1 – 1 = Md,2 – 2. We know that  Ka(d,2)  is  diregular.  Thus all  arcs which are previously going to u and now are going to vertex v implies that the in-degree of v is  larger  than d.  Consequently,  digraph G’ is not in-regular which means that digraph G’ is not diregular.Figure  3  illustrates  the  digraph  of  out-degree  4,  diameter  2  and  order  19  that  is constructed  from  Kautz  digraph  Ka(4,2)  as shown in Figure 2 by deleting vertex 12 and reconnecting its incoming arcs to vertex 02.     Figure 3. Non diregular digraph of order 19.CONCLUSIONWe  conclude  this  paper  with  summary  of known results  on the diregularity  of  digraphs which is updated from (Comellas  & Fiol 1990) as presented in Table 1 and research direction for future works. Jurnal ILMU DASAR Vol. 12 No. 1, Januari 2011 : 1-5 5Tabel 1. Diregularity of digraph out-degree  d, diameter k and order n.d k n Diregularity1 ≥ 1 M1,k Only diregular≥ 1 1 Md,1 Only diregular≥ 2 ≥ 2 Md,k-1 Only diregular2,3 2 Md,2-2 Diregular  &  non diregular2 ≥ 3 M2,k-2 Only diregular≥ 4 2 Md,2-2 Non  diregular  (the result),  diregular (unkown)≥ 3 ≥ 3 Md,k-2 Not kwonTable 1 shows that the diregularity of digraphs whose order are two less than the Moore bound (n =  Md,k  – 2) are still unknown, in particular for out-degree d ≥ 3 and diameter k ≥ 3.REFERENCESBaskoro ET,  Miller M, Siran J & Sutton M. 2005. Complete  characterisation  of  almost  Moore digraphs  of  degree  three.  Journal  of  Graph Theory. 48. Issue 2:112-126.Baskoro ET,  Miller  M & Plesnik J.  1998.  On the structure  of  digraphs  with  order  close  to  the Moore  bound.  Graphs  and  Combinatorics. 14 (2): 109 – 119.Bridges WG & Toueg S. 1980. On the impossibility of   directed  Moore  graphs.  Journal  of Combinatorial Theory. Series B 29: 339-341.Comellas F  &  Fiol  MA.  1990.  Using  simulated annealing  to  design  interconnection  networks. Technical  report DMAT-05-0290.  Universitat Politecnica de Catalunya. Presented at the  Fifth SIAM  Conference  on  Discrete  Mathematics. Atlanta.Dafik. 2007.  Structural  Properties  and Labeling of Graphs.  Ph.D  Thesis. University  of  Ballarat Australia.Hoffman AJ  &  Singleton  R.  1960.  On  Moore Graphs  with  Diameter  2  and  3.  IBM  J.  Res.  Develop. 4: 497 – 504.Fiol MA,  Yebra  JLA  &  Alegre  I.  1984.  Line Digraph  Iterations  and  The  (d,k)  Problem  for Directed  Graphs.  IEEE  Transactions  on Computers C-33: 400-403.Miller M & Fris I. 1992. Maximum Order Digraphs for Diameter 2 or Degree 2. Pullman volume of  Graphs  dan  Matrices,  Lecture  Notes  in  Pure dan Applied Mathematics 139: 269-278.Miller M & Siran J. 2001 Digraphs of Degree Two which Miss the Moore Bound by Two. Discrete Mathematics. 226:269-280.Miller  M,  Gimbert  J,  Siran  J  &  Slamin.  2000. Almost Moore Digraphs are Diregular. Discrete  Mathematics. 218 No.1-3: 265-270.Miller  M  &   Siran  J.  2005.  Moore  Graphs  and Beyond:  A  Survey  of  the  Degree/Diameter Problem. Electron. J. Combin. 1-61, DS14. Miller M & Slamin. 2000. On the Monotonicity of Minimum Diameter with Respect to Order and Maximum  Out-degree.  Proceedings  of  COCOON  2000,  Lecture  Notes  in  Computer  Science 1858  (D.-Z.  Du  P,  Eades  V,  Estivill-Castro, X Lin (eds.)), 193-201.Plesnik J  &  Znam  S.  1974.  Strongly  Geodetic Directed Graphs,  Acta F.R.N. Univ.  Comen. – Mathematica. XXIX: 29-34.Slamin & Miller M. 2000. Diregularity of Digraphs Close  to  Moore  bound.  Prosiding  Konperensi  Nasional Matematika X. MIHMI.6(5): 185-192. Slamin, Baskoro ET & Miller M. 2001. Diregularity of Digraphs of Out-degree Three and Order Two Less than Moore bound. Proceedings of Twelfth  Australian  Workshop  on  Combinatorial  Algorithms,164-173.

On The Existence of Non-Diregular Digraphs of Order Two Less than the Moore Bound

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On The Existence of Non-Diregular Digraphs of Order Two Less than the Moore Bound

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