A quartic vertex-transitive graph is a quartic graph that is vertex transitive. Read and Wilson (1988, pp. 164-166) enumerate all connected quartic vertex-transitive graphs on 19 and fewer nodes, some of which are illustrated above.
The quartic symmetric graphs are a special case of the quartic vertex-transitive graphs (i.e., those that are also edge-transitive).
Classes of connected quartic vertex-transitive graphs include the antiprism graphs. Specific cases are summarized in the following table. In particular, Qt31 can be constructed as the distance-3 graph of the Heawood graph or as the Levi graph of the biplane on 7 points (DistanceRegular.org). It is also a distance-regular graph with intersection array that is also distance-transitive.
vertices | id | graph |
5 | Qt1 | pentatope graph |
6 | Qt2 | octahedral graph |
8 | Qt5 | (2,4)-rook graph |
8 | Qt6 | complete bipartite graph |
9 | Qt9 | generalized quadrangle |
10 | Qt11 | crown graph |
12 | Qt20 | cuboctahedral graph |
14 | Qt31 | Heawood graph distance-3 graph |
15 | Qt39 | Petersen line graph |
16 | Qt51 | tesseract graph |
18 | Qt66 | Bouwer graph |