Graph Distance
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.
REFERENCES
Buckley, F. and Harary, F. Distance in Graphs. Redwood City, CA: Addison-Wesley, 1990.Diestel, R. Graph Theory, 3rd ed. New York: Springer-Verlag, p. 8, 1997.
Wilson, R. J. Introduction to Graph Theory, 3rd ed. New York: Longman, p. 30, 1985.
