 TOPICS  # Quartic Graph A quartic graph is a graph which is 4-regular. The unique quartic graph on five nodes is the complete graph , and the unique quartic graph on six nodes is the octahedral graph. There are two quartic graphs on seven nodes, one of which is the circulant graph . A number of the Archimedean solids have skeletons that are quartic.

The numbers of connected quartic graphs on , 2, ... nodes are 0, 0, 0, 0, 1, 1, 2, 6, 16, 59, ... (OEIS A006820), the numbers of not necessarily connected quartic graphs are 0, 0, 0, 0, 1, 1, 2, 6, 16, 60, ... (OEIS A033301), and the numbers of disconnected quartic graphs for , 11, ... are 1, 1, 3, 8, 25, 88, ... (OEIS A033483; Read and Wilson 1998).

The following table gives a list of some named quartic graphs.

 graph nodes symmetric pentatope graph 5 yes octahedral graph 6 yes complete bipartite graph 8 yes (2,4)-rook graph 8 no generalized quadrangle 9 yes 5-crown graph 10 yes 4-Andrásfai graph 11 no Chvátal graph 12 no cuboctahedral graph 12 yes 13-cyclotomic graph 13 yes Hoffman graph 16 no tesseract graph 16 yes Robertson graph 19 no Folkman graph 20 no Brinkmann graph 21 no generalized hexagon 21 yes rolling cube graph 24 yes small rhombicuboctahedral graph 24 no 25-Grünbaum graph 25 25 no (4,6)-cage graph 26 yes Doyle graph 27 yes icosidodecahedral graph 30 yes 4-odd graph 35 yes generalized octagon 45 yes Harborth graph 52 no small rhombicosidodecahedral graph 60 no (8,8)-fiveleaper graph 64 no (4,7)-cage graph 67 no Meredith graph 70 no (7,3)-bipartite Kneser graph 70 yes (4,8)-cage graph 80 no 5-permutation star graph 120 yes generalized dodecagon 189 no 120-cell graph 600 yes

120-Cell, Complete Graph, Cubic Graph, Cuboctahedral Graph, Doyle Graph, Grünbaum Graphs, Octahedral Graph, Quartic Symmetric Graph, Quintic Graph, Regular Graph

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## References

Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 648, 1996.Faradzev, I. A. "Constructive Enumeration of Combinatorial Objects." In Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Paris: Centre Nat. Recherche Scient., pp. 131-135, 1978.Meringer, M. "Connected Regular Graphs." http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html#CRG.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998.Sloane, N. J. A. Sequences A006820/M1617, A033301, and A033483 in "The On-Line Encyclopedia of Integer Sequences."

Quartic Graph

## Cite this as:

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