Addressing graph products and distance-regular graphs

Graham and Pollak showed that the vertices of any connected graph G can be assigned t-tuples with entries in {0,a,b}, called addresses, such that the distance in G between any two vertices equals the number of positions in their addresses where one of the addresses equals a and the other equals b. In this paper, we are interested in determining the minimum value of such t for various families of graphs. We develop two ways to obtain this value for the Hamming graphs and present a lower bound for the triangular graphs.