A bishop graph is a graph formed from possible moves of a bishop chess piece, which may make diagonal moves of any length on a chessboard (or any other board). To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable bishop moves are considered edges.
Because bishops starting on squares of one color and moving diagonally always remain on the same color squares, all bishop graphs are disconnected (except for the trivial
singleton graph on a 
board which is trivially connected).
Special cases are summarized in the following table.
 -bishop graph | graph |
 |  -empty
graph  |
 | 2  -path
graphs  |
The black/white components of an 
-bishop graph (i.e., white
bishop graph and black bishop graph) are
isomorphic iff 
and 
are not both odd.
Closed formulas for the numbers 
of 
-graph cycles of 
are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
and
 |
(3)
|
for 
, 7, ..., the last of which is due
to Perepechko and Voropaev.
S. Wagon (pers. comm., Aug. 17, 2012) showed that the 
-white bishop graph B(m,n) is Hamiltonian
for 
and when 
and 
, and nonhamiltonian for 
and the trivial cases 
or 1.
All bishop graphs are perfect.
