The distance
between two vertices
and
of a finite graph is the minimum length of the paths connecting them (i.e., the length
of a graph geodesic). If no such path exists (i.e.,
if the vertices lie in different connected components), then the distance is set
equal to .
In a grid graph the distance between two vertices is
the sum of the "vertical" and the "horizontal" distances (right
figure above).

The matrix
consisting of all distances from vertex to vertex is known as the all-pairs shortest path matrix, or more
simply, the graph distance matrix.