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# Quartic Vertex-Transitive Graph

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

Cubic Vertex-Transitive Graph, Quartic Graph, Quartic Symmetric Graph, Vertex-Transitive Graph

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## References

DistanceRegular.org. "Distance-3 Graph of Heawood Graph Incidence Graph of Biplane on 7 Points." http://www.distanceregular.org/graphs/heawood-dist3.html.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, pp. 164-166, 1998.

## Referenced on Wolfram|Alpha

Quartic Vertex-Transitive Graph

## Cite this as:

Weisstein, Eric W. "Quartic Vertex-Transitive Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/QuarticVertex-TransitiveGraph.html