The Degree Diameter Problem for General Graphs

Introduction
The degree/diameter problem for general graphs can be stated as follows:

Given natural numbers d and k, find the largest possible number N(d,k) of vertices in a graph of maximum degree d and diameter k.

In attempting to settle the values of N(d,k), research activities in this problem have follow the following two directions:


 * Increasing the lower bounds for N(d,k) by constructing ever larger graphs.

whose order is close to the Moore bounds M(d,k)=(d(d-1)k-2)(d-2)-1.
 * Lowering and/or setting upper bounds for N(d,k) by proving the non-existence of graphs

Increasing the lower bounds for N(d,k)
In the quest for the largest known graphs many innovative approaches have been suggested. In a wide spectrum, we can classify these approaches into general (those producing graphs for many combinations of the degree and the diameter) and ad hoc (those devised specifically for producing graphs for few combinations of the degree and the diameter). Among the former, we have the constructions of De Bruijn graphs and Kautz graphs, while among the latter, we have the star product, the voltage assigment technique and graph compunding. For information on the state-of -the-art of this research stream, the interested reader is referred to the survey by Miller and Širáň.

Below is the table of the largest known graphs (as of September 2009) in the undirected degree diameter problem for graphs of degree at most 3 ≤ d ≤ 20 and diameter 2 ≤ k ≤ 10. Only a few of the graphs in this table are known to be optimal (marked in bold), and thus, finding a larger graph that is closer in order (in terms of the size of the vertex set) to the Moore bound is considered an open problem. Some general constructions are known for values of d and k outside the range shown in the table.

Table of the orders of the largest known graphs for the undirected degree diameter problem
The following table is the key to the colors in the table presented above:

Lowering and/or setting upper bounds for N(d,k)
As the Moore bound cannot be reached in general, some theoretical work has been done to determine the lowest possible upper bounds. In this direction reserachers have been interested in graphs of maximum degree d, diameter k and order M(d,k)-&delta; for small &delta;. The parameter &delta; is called the defect. Such graphs are called (d,k,-&delta;)-graphs.

For &delta;=1 the only (d,k,-1)-graphs are the cycles on 2k vertices. Erd&ouml;s, Fajtlowitcz and Hoffman, who proved the non-existence of (d,2,-1)-graphs for d&ne;3. Then, Bannai and Ito, and also independently, Kurosawa and Tsujii, proved the non-existence of (d,k,-1)-graphs for d&ge;3 and k&ge;3.

For &delta;=2, the (2,k,-2)-graphs are the cycles on 2k-1. Considering d&ge;3, only five graphs are known at present. Elspas found the unique (4,2,-2)-graph and the unique (5,2,-2)-graph, and credited Green with producing the unique (3,3,-2)-graph. The other graphs are two non-isomorphic (3,2,-2)-graphs.

When &delta;=2, d&ge;3 and k&ge;3, not much is known about the existence or otherwise of (d,k,-2)-graphs. In this context some known outcomes include the non-existence of (3,k,-2)-graphs with k&ge;4 by Leif Jorgensen, the non-existence of (4,k,-2)-graphs with k&ge;3 by Mirka Miller and Rino Simanjuntak, some structural properties of (5,k,-2)-graphs with k&ge;3 by Guillermo Pineda-Villavicencio and Mirka Miller, the obtaining of several necessary conditions for the existence of (d,2,-2)-graphs with d&ge;3 by Mirka Miller, Minh Nguyen and Guillermo Pineda-Villavicencio, and the non-existence of (d,2,-2)-graphs for 5<d<50 by Jose Conde and Joan Gimbert.

For the case of &delta;&ge;3 only a few works are known at present: the non-existence of (3,4,-4)-graphs by Leif Jorgensen; the complete catalogue of (3,k,-4)-graphs with k&ge;2 by Guillermo Pineda-Villavicencio and Mirka Miller by proving the non-existence of (3,k,-4)-graphs with k&ge;5, the settlement of N(3,4)=M(3,4)=38 by Buset; and the obtaining of N(6,2)=M(6,2)-5=32 by Molodtsov. For more information, check the corresponding papers, and the survey by Miller and Širáň.

Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs
The following table is the key to the colors in the table presented above: