The resistance distance rij between two vertices vi and t,,j of a (connected, molecular) graph G is equal to the resistance between the respective two points of all electrical network, constructed so as to correspond to G, such that the resistance of ally two adjacent points is unity. We show how the matrix elements rij call be expressed in terms of the Laplacian eigenvalues and eigenvectors of G. In addition, we determine certain properties of the resistance matrix R=||rij||.