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
(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.