Knight's graph

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In graph theory, a knight's tour graph is a graph that represents all legal moves of the knight piece on a chessboard where each vertex represents a square on a chessboard and each edge is a legal move. More specifically, an $n \times m$ knight's tour graph is a knight's tour graph of an $n \times m$ chessboard.

For a $n \times m$ knight's tour graph the total number of vertices is simply $n m$ .

For a $n \times n$ knight's tour graph the total number of vertices is simply $n 2$ and the total number of edges is $4(n - 2)(n - 1)$ . Additionally, the number of edges for various $n$ is identified as A033996 in the On-Line Encyclopedia of Integer Sequences.