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Does this graph have a special name?

The vertices are arranged on a square square grid with a side length of $n$ and each inner vertices has an edge to its 8 neighbors.

And what about a similar graph with vertices arranged on a $n \times m$ rectangular grid?

Ruzzle™ graph

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it looks like the union of Cartesian product and the categorical product of two paths – hbm Dec 20 '13 at 21:50
up vote 6 down vote accepted

This is the King's graph. For all your Kingly needs.

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+1 'For all your Kingly needs' :) – Link Dec 20 '13 at 22:23

In terms of graph products the $n \times m$ grid will be the strong product of the path graphs $P_n$ and $P_m$, usually written as $P_n \boxtimes P_m$.

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