The 
-triangular
honeycomb king graph, called the hex king graph by Wagon (2014), is a graph
consisting of vertices in a triangular
honeycomb board with 
vertices along each side,
where vertices are connected by an graph
edge if they are adjacent along a horizontal, 
, or 
line of the chessboard (DeMaio and Tran 2013, Wagon
2014). It is denoted 
by DeMaio and Tran (2013) and 
by Wagon (2014). The graphs for
and 4 are illustrated above.
As is clear from the diagrams, the 
-triangular king graph is isomorphic to the triangular
grid graph 
of Wagon (2014) and the 
-triangular grid graph
using the indexing convention of West (2000).
Triangular honeycomb king graphs are apex, bridgeless, connected, Eulerian, Hamiltonian, linklessly embeddable, map, matchstick, planar, projective planar, quadratically embeddable, rigid, traceable, triangular grids, uniquely colorable, unit-distance, and weakly perfect.
Triangular honeycomb king graphs are implemented in the Wolfram Language as GraphData[
"TriangularHoneycombKing",
n
].
