8 - Symmetry

Summary

Prologue

I think that I shall never see

A poem showin' symmetry.

Anon.

The Petersen graph has exactly 120 automorphisms. All of its vertices are the same in the sense that any vertex can be mapped into any other by an automorphism (in fact by exactly 12 automorphisms). As we saw in Chapter 6 it has much more symmetry than this. All of its 120 paths of length 3 are the same. In fact there is exactly one automorphism which maps any one such path into any other. The Petersen graph is a Tutte graph since the constraint for the s–transitivity of a graph G given by s ≤ [½(γ(G) + 2)] is satisfied as an equality when G = P. In this sense P is as symmetric as it can be.

The Petersen graph has the property that any two pairs of vertices which are the same distance apart are also the same in the sense above. We say P is distance transitive. More precisely if u, v, x, y ∈ VP and d(u, v) = d(x, y) then there is an automorphism of P which maps u to x and v to y. Very surprisingly there are only twelve finite connected cubic distance transitive graphs.

The automorphism group of P acts primitively on its vertices. Roughly speaking this means that the automorphism group acts transitively on the vertex set and there is no k-subset of vertices (2 ≤ k < |VP|) which always stays together under the action of the automorphism group.