The Degree Diameter Problem for Vertex Symmetric Digraphs

Introduction
The degree/diameter problem for vertex-transitive digraphs can be stated as follows:

Given natural numbers d and k, find the largest possible number DNvt(d,k) of vertices in a vertex-transitive digraph of maximum out-degree d and diameter k''.

There are no better upper bounds for DNvt(d,k) than the very general directed Moore bounds DM(d,k)=(dk+1-1)(d-1)-1.

Therefore, in attempting to settle the values of DNvt(d,k), research activities in this problem follow the next two directions:


 * Increasing the lower bounds for DNvt(d,k) by constructing ever larger vertex-transitive digraphs.


 * Lowering and/or setting upper bounds for DNvt(d,k) by proving the non-existence of vertex-transitive digraphs whose order is close to the directed Moore bounds DM(d,k).

Increasing the lower bounds for DNvt(d,k)
Below is the table of the largest known digraphs (as of October 2008) in the degree diameter problem for digraphs of out-degree at most 3 ≤ d ≤ 16 and diameter 2 ≤ k ≤ 10. Only a few of the vertex-transitive digraphs in this table are known to be optimal (marked in bold), and thus, finding a larger vertex-transitive digraph whose order (in terms of the order of the vertex set) is close to the directed Moore bound is considered an open problem.

Table of the orders of the largest known vertex-symmetric graphs for the directed degree diameter problem
Digraphs in bold are optimal.

The following table is the key to the colors in the table presented above:

Lowering and/or setting upper bounds for DNvt(d,k)
With the exception of the studies done for the general digraphs, no study on this research area has been identified.