Editing Symmetric graph (section)

===Cubic symmetric graphs===
Combining the symmetry condition with the restriction that graphs be [[Cubic graph|cubic]] (i.e. all vertices have degree 3) yields quite a strong condition, and such graphs are rare enough to be listed. They all have an even number of vertices.  The '''Foster census''' and its extensions provide such lists.<ref>[[Marston Conder]], ''[http://www.math.auckland.ac.nz/~conder/preprints/cubic768.ps Trivalent symmetric graphs on up to 768 vertices],'' J. Combin. Math. Combin. Comput, vol. 20, pp. 41&ndash;63</ref> The Foster census was begun in the 1930s by [[R. M. Foster|Ronald M. Foster]] while he was employed by [[Bell Labs]],<ref>Foster, R. M. "Geometrical Circuits of Electrical Networks." ''[[Transactions of the American Institute of Electrical Engineers]]'' '''51''', 309&ndash;317, 1932.</ref> and in 1988 (when Foster was 92<ref name="biggs"/>) the then current Foster census (listing all cubic symmetric graphs up to 512 vertices) was published in book form.<ref>"The Foster Census: R.M. Foster's Census of Connected Symmetric Trivalent Graphs", by Ronald M. Foster, I.Z. Bouwer, W.W. Chernoff, B. Monson and Z. Star (1988) {{isbn|0-919611-19-2}}</ref>  The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices<ref>Biggs, p. 148</ref><ref name="F26A">Weisstein, Eric W., "[http://mathworld.wolfram.com/CubicSymmetricGraph.html Cubic Symmetric Graph]", from Wolfram MathWorld.</ref> (ten of these are also [[Distance-transitive graph|distance-transitive]]; the exceptions are as indicated):

{| class="wikitable"
|-
!'''Vertices''' !! '''[[Diameter (graph theory)|Diameter]]''' !! '''[[Girth (graph theory)|Girth]]''' !! '''Graph''' !! '''Notes'''
|-
|4 || 1 || 3 || The [[complete graph]] ''K''<sub>4</sub> || distance-transitive, 2-arc-transitive
|-
|6 || 2 || 4 || The [[complete bipartite graph]] ''K''<sub>3,3</sub> || distance-transitive, 3-arc-transitive
|-
|8 || 3 || 4 || The vertices and edges of the [[cube]] || distance-transitive, 2-arc-transitive
|-
|10 || 2 || 5 || The [[Petersen graph]] || distance-transitive, 3-arc-transitive
|-
|14 || 3 || 6 || The [[Heawood graph]] || distance-transitive, 4-arc-transitive
|-
|16 || 4 || 6 || The [[Möbius–Kantor graph]] || 2-arc-transitive
|-
|18 || 4 || 6 || The [[Pappus graph]] || distance-transitive, 3-arc-transitive
|-
|20 || 5 || 5 || The vertices and edges of the [[dodecahedron]] || distance-transitive, 2-arc-transitive
|-
|20 || 5 || 6 || The [[Desargues graph]] || distance-transitive, 3-arc-transitive
|-
|24 || 4 || 6 || The [[Nauru graph]] (the [[generalized Petersen graph]] G(12,5)) || 2-arc-transitive
|-
|26 || 5 || 6 || The [[F26A graph]]<ref name="F26A"/> || 1-arc-transitive
|-
|28 || 4 || 7 || The [[Coxeter graph]] || distance-transitive, 3-arc-transitive
|-
|30 || 4 || 8 || The [[Tutte–Coxeter graph]] || distance-transitive, 5-arc-transitive
|}

Other well known cubic symmetric graphs are the [[Dyck graph]], the [[Foster graph]] and the [[Biggs&ndash;Smith graph]].  The ten distance-transitive graphs listed above, together with the [[Foster graph]] and the [[Biggs&ndash;Smith graph]], are the only cubic distance-transitive graphs.