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.