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==Introduction==

The '''degree/diameter problem for arc-transitive graphs''' can be stated as follows:

''Given natural numbers ''d'' and ''k'', find the largest possible number ''N<sup>at</sup>(d,k)'' of vertices in an [http://en.wikipedia.org/wiki/Arc-transitive_graph arc-transitive graph] of maximum degree ''d'' and diameter ''k''.''

There are no better upper bounds for ''N<sup>at</sup>(d,k)'' than the very general ''Moore bounds'' ''M(d,k)=d((d-1)<sup>k</sup>-2)(d-2)<sup>-1</sup>''. 

Therefore, in attempting to settle the values of ''N<sup>at</sup>(d,k)'', research activities in this problem follow the next two directions:

* Increasing the lower bounds for ''N<sup>at</sup>(d,k)'' by constructing ever larger graphs.

* Lowering and/or setting upper bounds for ''N<sup>at</sup>(d,k)'' by proving the non-existence of arc-transitive graphs whose order is close to the Moore bounds ''M(d,k)''.

==Increasing the lower bounds for N<sup>at</sup>(d,k)==

With the exception of the graphs obtained by Conder, no study has been identified in this reserach area.

Below is the '''unfinished''' table of the largest known [http://en.wikipedia.org/wiki/Arc-transitive_graph arc-transitive graphs] in the undirected [[The Degree/Diameter Problem | degree diameter problem]] for arc-transitive graphs of [http://en.wikipedia.org/wiki/Degree_(graph_theory) degree] at most 3&nbsp;≤&nbsp;''d''&nbsp;≤&nbsp;20 and [http://en.wikipedia.org/wiki/Distance_(graph_theory) diameter] 2&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;10. '''Work in progress.'''

===Table of the orders of the largest known arc-transitive graphs for the undirected degree diameter problem===

Graphs in bold are known to be optimal. For each entry in the table we have the order of the graph and the largest 
value of ''r'' for which the known graph has ''r''-arc-transitive automorphism group.  (In some cases, where more 
than one graph exists, there can be two or more possibilities for this value of ''r''.) 

<center> 
{| border="1"
| '''<math>d</math>\<math>k</math>'''||  '''2''' ||  '''3''' ||  '''4'''|| '''5''' ||  '''6''' || '''7''' ||  '''8''' ||  '''9''' ||  '''10''' 
|-
| '''3'''
| align="center" |
{| border="2" style="background-color: red;" 
| '''10'''
|-
| 3
|}
| align="center" |
{| border="2" style="background-color: red;"
| '''14'''
|-
| 4
|}
| align="center" |
{| border="2" style="background-color: red;"
| '''30'''
|-
| 5
|}
| align="center" |
{| border="2" style="background-color: red;" 
| '''60'''
|-
| 2
|}
| align="center" |
{| border="2" style="background-color: red;" 
| '''64'''
|-
| 2
|}
| align="center" |
{| border="2" style="background-color: red;" 
| '''168'''
|-
| 1,2
|}
| align="center" |
{| border="2" style="background-color: red;" 
| '''234'''
|-
| 5
|}
| align="center" |
{| border="2" style="background-color: red;"
| '''364'''
|-
| 2
|}
| align="center" |
{| border="2" style="background-color: red;" 
| 1250
|-
| 3
|}
|-
| '''4'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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|-
| '''5'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
|}
|-
| '''6'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
|}
|-
| '''7'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
| '''8'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
| '''9'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
| '''10'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
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| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
| '''11'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
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| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
| '''12'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
| '''13'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
| '''14'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
| '''15'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
| '''16'''
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
| align="center" |
{| border="2" style="background:#ABCDEF;" 
| Size
|-
| r
|}
|-
|}
</center>

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

<center>
{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''Details'''
|-
|style="background-color: red; text-align: center;" | * || Graphs found by [[Marston Conder]].
|-
|style="background-color: blue; text-align: center;" | * || . 
|-
|style="background-color: #999900; text-align: center;" | * || .
|-
|style="background-color: #336666; text-align: center;" | * || . 
|-
|style="background-color: #ffff00; text-align: center;" | * || .
|-
|}
</center>

==Lowering and/or setting upper bounds for N<sup>at</sup>(d,k)==

No study has been identified in this reserach area.

==References==


==External links==
* [http://www.math.auckland.ac.nz/~conder/symmcubic2048list.txt Marston Conder's data] listing the complete set of graphs found and their description.




[[Category:The Degree/Diameter Problem]]