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)

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

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 .

See also 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." Referenced on Wolfram|Alpha Antiprism Graph
Cite this as:
Weisstein, Eric W. "Antiprism Graph."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/AntiprismGraph.html

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