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.
For a connected graph, summing the distances from a fixed vertex to all vertices gives the vertex's vertex transmission. Equivalently, vertex transmissions are the row sums of the graph distance matrix, and half the sum of all vertex transmissions is the Wiener index.
