DOCSLIB.ORG

Moore Graphs and Beyond: a Survey of the Degree/Diameter Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Moore Graphs and Beyond: a Survey of the Degree/Diameter Problem Moore graphs and beyond: A survey of the degree/diameter problem Mirka Miller School of Information Technology and Mathematical Sciences University of Ballarat, Ballarat, Australia [email protected] Jozef Sir´ˇ aˇn Department of Mathematics University of Auckland, Auckland, New Zealand [email protected] Submitted: Dec 4, 2002; Accepted: Nov 18, 2005; Published: Dec 5, 2005 Mathematics Subject Classifications: 05C88, 05C89 Abstract The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds { called Moore bounds { for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maxi- mum possible number of vertices, given the other two parameters, and thus attack- ing the degree/diameter problem `from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem `from below'. This sur- vey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand. the electronic journal of combinatorics (2005), #DS14 1 Contents 1 Introduction 3 2 Part 1: Undirected graphs 5 2.1Mooregraphs.................................. 5 2.2GraphsoforderclosetoMoorebound.................... 8 2.3Constructionsoflargegraphs......................... 11 2.3.1 Generaloverview............................ 11 2.3.2 Star product and compounding .................... 13 2.3.3 Tablesoflargegraphs......................... 15 2.4Restrictedversionsofthedegree/diameterproblem............. 16 2.4.1 Vertex-transitivegraphs........................ 16 2.4.2 Cayleygraphs.............................. 18 2.4.3 AbelianCayleygraphs......................... 20 2.4.4 Bipartitegraphs............................. 22 2.4.5 Graphsonsurfaces........................... 22 2.5Relatedtopics.................................. 23 2.5.1 Girth................................... 24 2.5.2 Connectivity............................... 25 3 Part 2: Directed graphs 27 3.1Mooredigraphs................................. 27 3.2DigraphsoforderclosetoMoorebound................... 27 3.3DiregularityofdigraphsclosetoMoorebound................ 31 3.4Constructionsoflargedigraphs........................ 33 3.5Restrictedversionsofthedegree/diameterproblem............. 38 3.5.1 Vertex-transitivedigraphs....................... 38 3.5.2 Cayleydigraphs............................. 42 3.5.3 Digraphsonsurfaces.......................... 43 3.6Relatedtopics.................................. 44 3.6.1 Connectivity............................... 44 3.6.2 Otherrelatedproblems......................... 45 4 Conclusion 47 the electronic journal of combinatorics (2005), #DS14 2 1 Introduction The topology of a network (such as a telecommunications, multiprocessor, or local area network, to name just a few) is usually modelled by a graph in which vertices represent ‘nodes’ (stations or processors) while undirected or directed edges stand for ‘links’ or other types of connections. In the design of such networks, there are a number of features that must be taken into account. The most common ones, however, seem to be limitations on the vertex degrees and on the diameter. The network interpretation of the two parameters is obvious: The degree of a vertex is the number of connections attached to a node, while the diameter indicates the largest number of links that must be traversed in order to transmit a message between any two nodes. What is then the largest number of nodes in a network with a limited degree and diameter? If links are modelled by undirected edges, this leads to the • Degree/Diameter Problem: Given natural numbers ∆ and D, find the largest pos- sible number of vertices n∆;D in a graph of maximum degree ∆ and diameter ≤ D. The statement of the directed version of the problem differs only in that ‘degree’ is replaced by ‘out-degree’. We recall that the out-degree of a vertex in a digraph is the number of directed edges leaving the vertex. We thus arrive at the • (Directed) Degree/Diameter Problem: Given natural numbers d and k, find the largest possible number of vertices nd;k in a digraph of maximum out-degree d and diameter ≤ k. Research activities related to the degree/diameter problem fall into two main streams. On the one hand, there are proofs of non-existence of graphs or digraphs of order close to the general upper bounds, known as the Moore bounds. On the other hand, there is a great deal of activity in the constructions of large graphs or digraphs, furnishing better lower bounds on n∆;D (resp., nd;k).Since the treatments of the undirected and directed cases have been quite different, we divide further exposition into two parts. Part 1 deals with the undirected case and Part 2 with the directed one. We first discuss the existence of Moore graphs (Section 2.1) and Moore digraphs (Section 3.1). These are graphs and digraphs which attain the so called Moore bound, giving the theoretical maximum for the order of a graphs (resp., digraph) given diameter and maximum degree (resp., out-degree). Then we present known results on the existence of graphs (Section 2.2) and digraphs (Section 3.2) whose order is ‘close’ to the Moore bound, whenever the Moore bound cannot be attained. the electronic journal of combinatorics (2005), #DS14 3 The question of regularity of graphs close to the Moore bound is much more interesting for directed graphs than for undirected ones, and so we include a section (3.3) on this topic only in Part 2. The next two sections (2.3 and 3.4) are then devoted to the constructions of large graphs and digraphs. In Sections 2.4 and 3.5 we introduce and discuss several restricted versions of the degree/diameter problem for graphs and digraphs. Various related topics are listed in Sections 2.5 and 3.6. Finally, in the Conclusion, we present a short list of some of the interesting open problems in the area. the electronic journal of combinatorics (2005), #DS14 4 2 Part 1: Undirected graphs 2.1 Moore graphs There is a straightforward upper bound on the largest possible order (i.e., the number of vertices) n∆;D of a graph G of maximum degree ∆ and diameter D. Trivially, if ∆ = 1 then D =1andn1;1 = 2; in what follows we therefore assume that ∆ ≥ 2. Let v be a vertex of the graph G and let ni, for 0 ≤ i ≤ D, be the number of vertices at distance i from v. Since a vertex at distance i ≥ 1fromv can be adjacent to at most ∆ − 1 vertices at distance i + 1 from v,wehaveni+1 ≤ (∆ − 1)ni, for all i such that i−1 1 ≤ i ≤ D − 1. With the help of n1 ≤ ∆, it follows that ni ≤ ∆(∆ − 1) , for 1 ≤ i ≤ D. Therefore, XD D−1 n∆;D = ni ≤ 1+∆+∆(∆− 1) + :::+∆(∆− 1) i=0 = 1+∆(1+(∆− 1) + :::+(∆− 1)D−1) ( − D− 1+∆ (∆ 1) 1 if ∆ > 2 = ∆−2 (1) 2D +1 if∆=2 The right-hand side of (1) is called the Moore bound and is denoted by M∆;D. The bound was named after E. F. Moore who first proposed the problem, as mentioned in [160]. A graph whose order is equal to the Moore bound M∆;D is called a Moore graph;sucha graph is necessarily regular of degree ∆. The study of Moore graphs was initiated by Hoffman and Singleton. Their pioneering paper [160] was devoted to Moore graphs of diameter 2 and 3. In the case of diameter D = 2, they proved that Moore graphs exist for ∆ = 2; 3; 7 and possibly 57 but for no other degrees, and that for the first three values of ∆ the graphs are unique. For D =3 they showed that the unique Moore graph is the heptagon (for ∆ = 2). The proofs exploit eigenvalues and eigenvectors of the adjacency matrix (and its principal submatrices) of graphs. It turns out that no Moore graphs exist for the parameters ∆ ≥ 3andD ≥ 3. This was shown by Damerell [95] by way of an application of his theory of distance-regular graphs to the classification of Moore graphs. An independent proof of this result was also given by Bannai and Ito [19]. Main results concerning Moore graphs can therefore be summed up as follows. Moore graphs for diameter D =1anddegree∆≥ 1 are the complete graphs K∆+1.For diameter D = 2, Moore graphs are the cycle C5 for degree ∆ = 2, the Petersen graph (see Fig. 1) for degree ∆ = 3, and the Hoffman-Singleton graph (see Fig. 2, drawn by Slamin [220]) for degree ∆ = 7. Finally, for diameter D ≥ 3 and degree ∆ = 2, Moore graphs are the cycles on 2D + 1 vertices C2D+1. the electronic journal of combinatorics (2005), #DS14 5 Figure 1: Petersen graph. Between the time of the publication of the Hoffman-Singleton paper (1960) and the time of the publication of the results by Bannai-Ito and Damerell (both in 1973), there were several related partial results published. For example, Friedman [134] showed that there are no Moore graphs for parameters (∆;D), where ∆ = 3; 4; 5; 6; 8and3<D≤ 300, except, possibly, for the pair (5; 7). He also showed that there are no Moore graphs with parameters (3;D), when D ≥ 3and2D + 1 is prime. Bos´ak [60] proved the nonexistence of Moore graphs of degree 3 and diameter D,3≤ D ≤ 8. For another contribution to nonexistence proofs, see also Plesn´ık [202]. A combinatorial proof that the Moore graph with ∆ = 7 and D = 2 is unique was given by James [167]. As an aside, a connection of Moore graphs with design theory was found by Benson and Losey [38], by embedding the Hoffman-Singleton graph in the projective plane PG(2; 52). Several other areas of research in graph theory turn out to be related or inspired by the theory of Moore graphs; examples include cages, antipodal graphs, Moore geometries, and Moore groups.
Load more

Recommended publications
  • Avoiding Rainbow Induced Subgraphs in Vertex-Colorings
    Avoiding rainbow induced subgraphs in vertex-colorings Maria Axenovich and Ryan Martin∗ Department of Mathematics, Iowa State University, Ames, IA 50011 [email protected], [email protected] Submitted: Feb 10, 2007; Accepted: Jan 4, 2008 Mathematics Subject Classification: 05C15, 05C55 Abstract For a fixed graph H on k vertices, and a graph G on at least k vertices, we write G −→ H if in any vertex-coloring of G with k colors, there is an induced subgraph isomorphic to H whose vertices have distinct colors. In other words, if G −→ H then a totally multicolored induced copy of H is unavoidable in any vertex-coloring of G with k colors. In this paper, we show that, with a few notable exceptions, for any graph H on k vertices and for any graph G which is not isomorphic to H, G 6−→ H. We explicitly describe all exceptional cases. This determines the induced vertex- anti-Ramsey number for all graphs and shows that totally multicolored induced subgraphs are, in most cases, easily avoidable. 1 Introduction Let G =(V, E) be a graph. Let c : V (G) → [k] be a vertex-coloring of G. We say that G is monochromatic under c if all vertices have the same color and we say that G is rainbow or totally multicolored under c if all vertices of G have distinct colors. The existence of a arXiv:1605.06172v1 [math.CO] 19 May 2016 graph forcing an induced monochromatic subgraph isomorphic to H is well known. The following bounds are due to Brown and R¨odl: Theorem 1 (Vertex-Induced Graph Ramsey Theorem [6]) For all graphs H, and all positive integers t there exists a graph Rt(H) such that if the vertices of Rt(H) are colored with t colors, then there is an induced subgraph of Rt(H) isomorphic to H which is monochromatic.
    [Show full text]
  • Strongly Regular Graphs
    MT5821 Advanced Combinatorics 1 Strongly regular graphs We introduce the subject of strongly regular graphs, and the techniques used to study them, with two famous examples. 1.1 The Friendship Theorem This theorem was proved by Erdos,˝ Renyi´ and Sos´ in the 1960s. We assume that friendship is an irreflexive and symmetric relation on a set of individuals: that is, nobody is his or her own friend, and if A is B’s friend then B is A’s friend. (It may be doubtful if these assumptions are valid in the age of social media – but we are doing mathematics, not sociology.) Theorem 1.1 In a finite society with the property that any two individuals have a unique common friend, there must be somebody who is everybody else’s friend. In other words, the configuration of friendships (where each individual is rep- resented by a dot, and friends are joined by a line) is as shown in the figure below: PP · PP · u P · u · " " BB " u · " B " " B · "B" bb T b ub u · T b T b b · T b T · T u PP T PP u P u u 1 1.2 Graphs The mathematical model for a structure of the type described in the Friendship Theorem is a graph. A simple graph (one “without loops or multiple edges”) can be regarded as a set carrying an irreflexive and symmetric binary relation. The points of the set are called the vertices of the graph, and a pair of points which are related is called an edge.
    [Show full text]
  • Dynamic Cage Survey
    Dynamic Cage Survey Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Robert Jajcay Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Department of Algebra Comenius University Bratislava, Slovakia [email protected] Submitted: May 22, 2008 Accepted: Sep 15, 2008 Version 1 published: Sep 29, 2008 (48 pages) Version 2 published: May 8, 2011 (54 pages) Version 3 published: July 26, 2013 (55 pages) Mathematics Subject Classifications: 05C35, 05C25 Abstract A(k; g)-cage is a k-regular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions. the electronic journal of combinatorics (2013), #DS16 1 Contents 1 Origins of the Problem 3 2 Known Cages 6 2.1 Small Examples . 6 2.1.1 (3,5)-Cage: Petersen Graph . 7 2.1.2 (3,6)-Cage: Heawood Graph . 7 2.1.3 (3,7)-Cage: McGee Graph . 7 2.1.4 (3,8)-Cage: Tutte-Coxeter Graph . 8 2.1.5 (3,9)-Cages . 8 2.1.6 (3,10)-Cages . 9 2.1.7 (3,11)-Cage: Balaban Graph . 9 2.1.8 (3,12)-Cage: Benson Graph . 9 2.1.9 (4,5)-Cage: Robertson Graph . 9 2.1.10 (5,5)-Cages .
    [Show full text]
  • On Total Regularity of Mixed Graphs with Order Close to the Moore Bound
    On total regularity of mixed graphs with order close to the Moore bound James Tuite∗ Grahame Erskine ∗ [email protected] [email protected] Abstract The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected edges and directed arcs. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter k, maximum undirected degree r and maximum directed out-degree z. It is also of interest to find smallest ≤ ≤ possible k-geodetic mixed graphs with minimum undirected degree r and minimum directed ≥ out-degree z. A simple counting argument reveals the existence of a natural bound, the ≥ Moore bound, on the order of such graphs; a graph that meets this limit is a mixed Moore graph. Mixed Moore graphs can exist only for k = 2 and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For k = 2, we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of L´opez and Miret. We also present partial results for larger k. We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one.
    [Show full text]
  • New Large Graphs with Given Degree and Diameter ∗
    New Large Graphs with Given Degree and Diameter ∗ F.Comellas and J.G´omez Departament de Matem`aticaAplicada i Telem`atica Universitat Polit`ecnicade Catalunya Abstract In this paper we give graphs with the largest known order for a given degree ∆ and diameter D. The graphs are constructed from Moore bipartite graphs by replacement of some vertices by adequate structures. The paper also contains the latest version of the (∆,D) table for graphs. 1 Introduction A question of special interest in Graph theory is the construction of graphs with an order as large as possible for a given maximum de- gree and diameter or (∆,D)-problem. This problem receives much attention due to its implications in the design of topologies for inter- connection networks and other questions such as the data alignment problem and the description of some cryptographic protocols. The (∆,D) problem for undirected graphs has been approached in different ways. It is possible to give bounds to the order of the graphs for a given maximum degree and diameter (see [5]). On the other hand, as the theoretical bounds are difficult to attain, most of the work deals with the construction of graphs, which for this given diameter and maximum degree have a number of vertices as close as possible to the theoretical bounds (see [4] for a review). Different techniques have been developed depending on the way graphs are generated and their parameters are calculated. Many (∆,D) ∗ Research supported in part by the Comisi´onInterministerial de Ciencia y Tec- nologia, Spain, under grant TIC90-0712 1 graphs correspond to Cayley graphs [6, 11, 7] and have been found by computer search.
    [Show full text]
  • Spectra of Graphs
    Spectra of graphs Andries E. Brouwer Willem H. Haemers 2 Contents 1 Graph spectrum 11 1.1 Matricesassociatedtoagraph . 11 1.2 Thespectrumofagraph ...................... 12 1.2.1 Characteristicpolynomial . 13 1.3 Thespectrumofanundirectedgraph . 13 1.3.1 Regulargraphs ........................ 13 1.3.2 Complements ......................... 14 1.3.3 Walks ............................. 14 1.3.4 Diameter ........................... 14 1.3.5 Spanningtrees ........................ 15 1.3.6 Bipartitegraphs ....................... 16 1.3.7 Connectedness ........................ 16 1.4 Spectrumofsomegraphs . 17 1.4.1 Thecompletegraph . 17 1.4.2 Thecompletebipartitegraph . 17 1.4.3 Thecycle ........................... 18 1.4.4 Thepath ........................... 18 1.4.5 Linegraphs .......................... 18 1.4.6 Cartesianproducts . 19 1.4.7 Kronecker products and bipartite double. 19 1.4.8 Strongproducts ....................... 19 1.4.9 Cayleygraphs......................... 20 1.5 Decompositions............................ 20 1.5.1 Decomposing K10 intoPetersengraphs . 20 1.5.2 Decomposing Kn into complete bipartite graphs . 20 1.6 Automorphisms ........................... 21 1.7 Algebraicconnectivity . 22 1.8 Cospectralgraphs .......................... 22 1.8.1 The4-cube .......................... 23 1.8.2 Seidelswitching. 23 1.8.3 Godsil-McKayswitching. 24 1.8.4 Reconstruction ........................ 24 1.9 Verysmallgraphs .......................... 24 1.10 Exercises ............................... 25 3 4 CONTENTS 2 Linear algebra 29 2.1
    [Show full text]
  • A Survey on the Missing Moore Graph ∗
    A survey on the missing Moore graph ∗ C. Dalf´o Departament de Matem`atica Universitat de Lleida Igualada (Barcelona), Catalonia [email protected] Abstract This is a survey on some known properties of the possible Moore graph (or graphs) Υ with degree 57 and diameter 2. Moreover, we give some new results about it, such as the following. When we consider the distance partition of Υ induced by a vertex subset C, the following graphs are distance-regular: The induced graph of the vertices at distance 1 from C when C is a set of 400 independent vertices; the induced graphs of the vertices at distance 2 from C when C is a vertex or an edge, and the line graph of Υ. Besides, Υ is an edge-distance-regular graph. Keywords: Moore graphs, Distance-regular graph, Outindependent graph, Spectral graph theory, Automorphisms. Mathematics Subject Classifications: 05C50; 05C25; 20B25; 20C15. 1 Introduction The Moore bound M(∆;D) is an upper bound on the largest possible number of vertices of a graph G with maximum degree ∆ and diameter D. Trivially, if ∆ = 1, then D = 1 and M(1; 1) = 2. If ∆ = 2, then M(2;D) = 2D + 1. For ∆ > 2, (∆ − 1)D − 1 M(∆;D) = 1 + ∆ : ∆ − 2 ∗This research is partially supported by the project 2017SGR1087 of the Agency for the Management of University and Research Grants (AGAUR) of the Government of Catalonia. This research has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agree- ment No 734922.
    [Show full text]
  • Distance-Regular Graphs Arxiv:1410.6294V2 [Math.CO]
    Distance-regular graphs∗ Edwin R. van Dam Jack H. Koolen Department of Econometrics and O.R. School of Mathematical Sciences Tilburg University University of Science and Technology of China The Netherlands and [email protected] Wu Wen-Tsun Key Laboratory of Mathematics of CAS Hefei, Anhui, 230026, China [email protected] Hajime Tanaka Research Center for Pure and Applied Mathematics Graduate School of Information Sciences Tohoku University Sendai 980-8579, Japan [email protected] Mathematics Subject Classifications: 05E30, 05Cxx, 05Exx Abstract This is a survey of distance-regular graphs. We present an introduction to distance- regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph `BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written. Keywords: Distance-regular graph; survey; association scheme; P -polynomial; Q- polynomial; geometric arXiv:1410.6294v2 [math.CO] 15 Apr 2016 ∗This version is published in the Electronic Journal of Combinatorics (2016), #DS22. 1 Contents 1 Introduction6 2 An introduction to distance-regular graphs8 2.1 Definition . .8 2.2 A few examples . .9 2.2.1 The complete graph . .9 2.2.2 The polygons . .9 2.2.3 The Petersen graph and other Odd graphs . .9 2.3 Which graphs are determined by their intersection array? . .9 2.4 Some combinatorial conditions for the intersection array . 11 2.5 The spectrum of eigenvalues and multiplicities . 12 2.6 Association schemes . 15 2.7 The Q-polynomial property .
    [Show full text]
  • Notes on Extremal Graph Theory
    Notes on Extremal Graph Theory Ryan Martin April 5, 2012 2 Contents 1 Prologue 7 1.1 Apologies . .7 1.2 Thanks . .7 2 Introduction to Extremal Graph Theory 9 2.1 Notation and terminology . 10 2.1.1 General mathematical notation . 10 2.1.2 Graph theory notation . 10 2.1.3 Some special graph families . 11 2.1.4 Useful bounds . 11 3 The Basics 13 3.1 K¨onig-Halltheorem . 13 3.1.4 Equivalent theorems . 15 3.2 Tutte's theorem . 17 3.3 Tur´an'stheorem . 18 3.4 K¨onigsberg . 20 3.5 Dirac's theorem . 20 3.6 The Hajnal-Szemer´editheorem . 21 3.7 Gems: The Hoffman-Singleton theorem . 26 4 Ramsey Theory 31 4.1 Basic Ramsey theory . 31 4.1.4 Infinite Ramsey theory . 32 4.2 Canonical Ramsey theory . 33 4.2.2 Infinite version . 34 4.2.4 Canonical Ramsey numbers . 35 5 The Power of Probability 37 5.1 Probability spaces . 37 5.1.1 Formal definitions . 37 5.1.2 Probability in a discrete setting . 38 5.1.3 Mean and variance . 38 5.1.5 Independence . 39 3 4 CONTENTS 5.1.6 Expectation . 39 5.2 Linearity of expectation . 40 5.2.2 A lower bound for diagonal Ramsey numbers . 40 5.2.4 Finding a dense bipartite subgraph . 41 5.2.6 Dominating sets . 42 5.3 Useful bounds . 43 5.4 Chernoff-Hoeffding bounds . 44 5.4.1 Independence . 44 5.4.3 A general Chernoff bound . 45 5.4.6 Binomial random variables .
    [Show full text]
  • 20 Apr 2020 Eigenvalues of Graphs and Spectral Moore Theorems
    Eigenvalues of graphs and spectral Moore theorems Sebastian M. Cioab˘a∗ April 21, 2020 Abstract In this paper, we describe some recent spectral Moore theorems related to determining the maximum order of a connected graph of given valency and second eigenvalue. We show how these spectral Moore theorems have applications in Alon-Boppana theorems for reg- ular graphs and in the classical degree-diameter/Moore problem. §1 Introduction Our graph theoretic notation is standard (see [4, 5]). Let Γ = (V, E) be an undirected graph with vertex set V and edge set E. Given u, v V , the distance d(u, v) equals the minimum length of a path between u∈and v if such a path exists or otherwise. If Γ is connected, then all the distances between its vertices are∞ finite and the diameter diam(Γ) of Γ is defined as the maximum of d(u, v), where the maximum is taken over all pairs u, v V . The pairwise distance and the diameter of a connected graph can be calc∈ ulated efficiently using breadth first search. The Moore or degree-diameter problem is a classical problem in combinatorics (see [31]). arXiv:2004.09221v1 [math.CO] 20 Apr 2020 Problem 1. Given r 3 and D 2, what is the maximum order nr,D of a connected r-regular graph≥ of diameter≥ D ? There is a well known upper bound for nr,D known as the Moore bound mr,D which is obtained as follows. If Γ is a connected r-regular graph of ∗Department of Mathematical Sciences, University of Delaware, Newark, DE 19716- 2553, USA.
    [Show full text]
  • Variations and Generalizations of Moore Graphs
    Variations and Generalizations of Moore Graphs. Iwont 2012, Bandung Leif K. Jørgensen Aalborg University Denmark The Petersen graph Moore Graphs An (undirected) graph with (maximum) degree ∆ and diameter D has or- der (number of vertices) at most M(∆;D) := 1 + ∆ + ∆(∆ − 1) + ∆(∆ − 1)2 + ::: + ∆(∆ − 1)D−1: A graph with with (minimum) degree ∆ and girth 2D + 1 has at least M(∆;D) vertices. M(∆;D) is called the Moore bound. A graph with exactly M(∆;D) vertices has maximum degree ∆ and diameter D if and only if it has minimum degree ∆ and girth 2D + 1. If these properties are satisfied then graph is called a Moore graph. Theorem Hoffman + Singleton 1960 If there exists a Moore graph with diameter D = 2 then the degree is either • ∆ = 2, unique Moore graph: the cycle of length 5, • ∆ = 3, unique Moore graph: the Petersen graph, • ∆ = 7, unique Moore graph: the Hoffman-Singleton graph, or • ∆ = 57, existence of Moore graph is unknown. Theorem Damerell 1973, Bannai + Ito 1973 A Moore graph with ∆ ≥ 3 and D ≥ 3 does not exist. Does there exist a Moore graph with D = 2 and ∆ = 57. Number of vertices is 3250. Theorem Macajˇ + Sirˇ a´nˇ 2010 A Moore graph of degree 57 has at most 375 automorphisms. Does a Moore graph of degree 57 contain the Petersen graph ? Bipartite Moore Graphs A graph with (minimum) degree ∆ and girth 2D has order at least D−1 MB(∆;D) := 2(1 + (∆ − 1) + (∆ − 1) + ::: + (∆ − 1) ): If a graph with (minimum) degree ∆ and girth 2D has order MB(∆;D) then it is regular and bipartite and it is said to be a bipartite Moore graph.
    [Show full text]
  • Second Neighbourhoods of Strongly Regular Graphs
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 103 (1992) 161-170 161 North-Holland Second neighbourhoods of strongly regular graphs A.D. Gardiner Department of Mathematics, University of Birmingham, Birmingham, UK, B 15 ZT? C.D. Godsil,* A.D. Hensel** and Gordon F. Royle* Department of Combinatorics and Optimization, University of Waterloo, Ont., Canada Received 18 January 1988 Revised 16 May 1989 Dedicated to Professor R.G. Stanton. Abstract Gardiner, A.D., CD. Godsil, A.D. Hensel and G.F. Royle, Second neighbourhoods of strongly regular graphs, Discrete Mathematics 103 (1992) 161-170. Some antipodal distance-regular graphs of diameter three arise as the graph induced by the vertices at distance two from a given vertex in a strongly regular graph. We show that if every vertex in a strongly regular graph G has this property, then G is the noncollinearity graph of a special type of semipartial geometry. As these semipartial geometries have all been classified, we obtain a list of the antipodal distance-regular graphs of diameter three that can arise in this way. 1. Introduction Distance-regular graphs may be divided into the two classes of primitive and imprimitive graphs. (Definitions for all terms may be found in the next section.) Primitive distance-regular graphs have been extensively studied [4], but much less is known about the imprimitive case. Smith [13] showed that imprimitive distance-regular graphs of valency at least three must either be antipodal or bipartite (or both). This work arose whilst studying antipodal distance-regular graphs of diameter three, which are necessarily covers of complete graphs.
    [Show full text]