The 
-crown graph for an integer 
is the graph with vertex set
 |
(1)
|
and edge set
 |
(2)
|
It is therefore equivalent to the complete bipartite graph 
with horizontal edges removed.
Note that the term "crown graph" has also been used to refer to a sunlet graph 
(e.g., Gallian 2018).
The 
-crown graph is isomorphic to the rook
complement graph 
(somewhat confusingly phrased as the complement of a 
grid by Brouwer et al. 1989, p. 222, Theorem
7.5.2, item (iii)), where 
denotes the graph Cartesian product. The
-crown graph is also isomorphic to a complete bipartite graph 
minus an independent
edge set (cf. Brouwer and Koolen 1999).
The crown graphs are distance-transitive (Brouwer et al. 1989, p. 222), and therefore also distance-regular. They are also Taylor graphs.
The line graph of the 
-crown graph is the arrangement
graph 
.
The 
-crown graph is isomorphic to the Haar
graph 
.
Other special cases are summarized in the following table.
The numbers of vertices in 
is 
, which the number of edges is 
.
The numbers of directed Hamiltonian cycles for 
, 4, 5, ... are given by 2, 12, 312,
9600, 416880, ... (OEIS A094047), which has
the beautiful closed form
 |
(3)
|
(M. Alekseyev, pers. comm., Feb. 10, 2008).
Closed formulas for the numbers 
of 
-graph cycles of 
are given by 
for 
odd and
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
(E. Weisstein, Nov. 16, 2014).
The independence polynomial of the 
-crown graph is
 |
(7)
|
which satisfies the recurrence equation
 |
(8)
|
