Quartic Symmetric Graph

Following the vertex-and-edge-transitive convention, a quartic symmetric graph is a regular graph of vertex degree 4 that is both vertex-transitive and edge-transitive. The following table gives the numbers of nonisomorphic quartic symmetric graphs on , 2, ... vertices, with separate counts for the connected and disconnected subclasses.

Some quartic symmetric graphs are illustrated above. The following table gives a selection of notable examples. Here denotes the Petersen graph, the line graph of a graph , and a perfect matching.

graph	nodes
pentatope graph	5
octahedral graph	6
complete bipartite graph	8
crown graph	10
cuboctahedral graph	12
Petersen line graph	15
tesseract graph	16
torus grid graph	25
Doyle graph	27
icosidodecahedral graph	30
torus grid graph	36
Bouwer graph	54

Among quartic symmetric graphs, the 54-vertex Bouwer graph constructed by Bouwer (1970) is half-arc-transitive. The smaller 27-vertex Doyle graph was independently discovered by Doyle (1976) and Holt (1981).

Among quartic symmetric graphs, Potočnik et al. (2013) enumerated the arc-transitive subclass through 640 vertices, comprising 4,820 connected graphs.