Hamming distances [in networks]

In the so-called loop switching method of routing messages in communications systems one lays out a network on an m-dimensional Boolean hypercube so that the distance on the hypercube (equal to Hamming distance) agrees with distance in the network. It is known that to achieve this exactly, m must be at the least the number of either positive or negative eigenvalues of the distance matrix for the network, and can need to be as much as n-1, where n is the total number of nodes.