Abstract:
The arc-type of a vertex-transitive graph is a partition of the graph's valency as a sum of the lengths of the orbits of a vertex-stabiliser on the neighbourhood of that vertex, with parentheses used in this partition to denote paired orbits. It has been shown by Conder, Pisanski and Zitnik that every arc-type except for 2 = 1+1 and 2 = (1+1) is realised by some vertex-transitive graph [8]. We extend their theorem by showing that every arc-type other than 1, 1+1 and (1+1) is realised by infinitely many connected finite Cayley graphs.