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 
. Equivalently, it is the cycle
power graph 
,
the graph square of the cycle
graph 
.
In turn, the graph square of the 
-antiprism graph is the circulant
graph 
and its graph cube is 
.
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)
|
For 
,
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.
