  Lipman et al. [ACM Transactions on Graphics 29 (3) (2010), 1--11] introduced
the concept of biharmonic distance to measure the distances between pairs of
points on a 3D surface. Biharmonic distance has some advantages over resistance
distance and geodesic distance in some realistic contexts. Nevertheless,
limited work has been done on the biharmonic distance in the discrete case. In
this paper, we give some characterizations of the biharmonic distance of a
graph. Some basic mathematical properties of biharmonic distance and biharmonic
index are established.
