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
an 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 bishop, 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
].
