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.
The graph square of is the circulant graph and its graph cube is .
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)
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or
(2)
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(Golin and Leung 2004; M. Alekseyev, pers. comm., Feb. 7, 2008), which has the closed-form solution
(3)
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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)
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where
(5)
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(6)
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The recurrence relations for the chromatic polynomial, independence polynomial, and matching polynomial are
(7)
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The 6-antiprism graph is cospectral with the quartic vertex-transitive graph Qt19, meaning neither is determined by spectrum.