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# Antiprism Graph

An antiprism graph is a graph corresponding to the skeleton of an antiprism. Antiprism graphs are therefore polyhedral and planar. The -antiprism graph has vertices and edges, and is isomorphic to the circulant graph . The 3-antiprism graph is also isomorphic to the octahedral graph.

Precomputed properties for antiprism graphs are implemented in the Wolfram Language as GraphData["Antiprism", n].

The numbers of directed Hamiltonian cycles for , 4, ... are 32, 58, 112, 220, 450, 938, 1982, ... (OEIS A124353), whose terms are given by the recurrence relation

 (1)

or

 (2)

(Golin and Leung 2004; M. Alekseyev, pers. comm., Feb. 7, 2008), which has the closed-form solution

 (3)

where , , and are the roots of .

The antiprism graphs are pancyclic. -antiprism graphs are nut graphs when is not divisible by 3.

The numbers of graph cycles on the -antiprism graph for , 4, ... are 63, 179, 523, ... (OEIS A077263), illustrated above for .

The -antiprism graph has chromatic polynomial

 (4)

where

 (5) (6)

The recurrence relations for the chromatic polynomial, independence polynomial, and matching polynomial are

 (7)

The 6-antiprism graph is cospectral with the quartic vertex-transitive graph Qt19, meaning neither is determined by spectrum.

Antiprism, Circulant Graph, Cospectral Graphs, Determined by Spectrum, Prism Graph

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## References

Golin, M. J. and Leung, Y. C. "Unhooking Circulant Graphs: a Combinatorial Method for Counting Spanning Trees and Other Parameters." In Graph-Theoretic Concepts in Computer Science. Revised Papers from the 30th International Workshop (WG 2004) Held in Bad Honnef, June 21-23, 2004 (Ed. J. Hromkovič, M. Nagl, and B. Westfechtel). Berlin: Springer-Verlag, pp. 296-307, 2004.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, p. 263 and 270, 1998.Sloane, N. J. A. Sequences A077263 and A124353 in "The On-Line Encyclopedia of Integer Sequences."

Antiprism Graph

## Cite this as:

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