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. More formally, the 
-crown graph is equivalent to 
minus a perfect matching
(cf. Brouwer and Koolen 1999, van Dam and Haemers 2003).
Note that the term "crown graph" has also been used to refer to a sunlet graph 
(e.g., Gallian 2018).
The 
-crowns used in the "crowning operation"
(Koester 1990) are unrelated to the crown graphs defined here. Repeated application
of that "crowning" operation to the Chiu graph
generates an infinite family of quartic planar edge-critical graphs with fractional
chromatic number 3 (Chiu et al. 2024).
The 
-crown graph is also 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 graphs are distance-transitive (Brouwer et al. 1989, p. 222), and therefore also distance-regular. They are also Taylor graphs, as well as determined by spectrum graphs (Proposition 9(ii) in van Dam and Haemers (2003)).
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)
|
