The 
-triangular
honeycomb bishop graph 
(DeMaio and Tran 2013), called the hex bishop graph and
denoted 
by Wagon (2014), is a graph consisting of vertices
on a triangular honeycomb board with
vertices along each side, where vertices
are connected by a graph edge if they lie on the same
or 
diagonal line of the chessboard (DeMaio and Tran 2013, Wagon 2014). The graphs
for 
and 4 are illustrated above.
Note the moves considered in this definition differ from those allowed by the bishop piece in Gliński's hexagonal chess (Gliński 1973).
Rather surprisingly, the 
-triangular honeycomb bishop graph is isomorphic to the 
black bishop graph (Wagon 2014). Other special
cases are summarized in the following table.
 | isomorphic graph |
| 1 | singleton graph  |
| 2 | path
graph  |
| 3 | cis-square with two triangles |
The 
-triangular
honeycomb bishop graph has vertex count and edge
count given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is a binomial coefficient.
Triangular honeycomb bishop graphs are black bishops, class 1, claw-free, connected, line, nongeometric, perfect, quadratically embeddable, traceable, and weakly perfect.
Triangular honeycomb bishop graphs are implemented in the Wolfram Language as GraphData[
"TriangularHoneycombBishop", n
].
