Is there a name for a graph very similar to a hypercube, but generalized from $2^d$ vertices to $k^d$? Alternatively, similar to a 2-dimensional grid, but generalized to higher dimensions?


In 2 dimensions, it would be a square grid with a side length of $k$. In 3 or more dimensions the graph would look like a subdivided hypercube, with internal vertices each having $2d$ neighbors.


Vertices can be given coordinates with base $k$ and length $d$, with adjacent vertices differing only by one in a single dimension.


Is there a proper name for this structure? I haven't found any discussion of it with the terms used above.



Proper name for hypergrid or subdivided hypercube?
Here is a (incomplete) list of names that people use for the object (with examples):


* "generalized grid graph" ([mathematica](http://mathworld.wolfram.com/GridGraph.html))
* "hypergrid graph" ([matlab](http://www.mathworks.com/matlabcentral/fileexchange/10922-matlabbgl/content/matlab_bgl/grid_graph.m))
* "multi-dimensional grid graph" ([BHS](http://arxiv.org/abs/0711.4151))


Perhaps "hypercubical grid graph" would also not be a bad choice.


The notation $[k]^n$ suggested by Imre Leader through Daniel Soltész makes perfect sense as the $k$-fold product of the path graph $[k]$.


