Addresses for graphs

The problem of labeling the vertices of an undirected, connected graph with binary
n
-tuple addresses is considered. These addresses are to have the property that if two vertices are a distance
k
apart in the graph then the Hamming distance between the corresponding addresses must be
kd
where
d
is a positive integer which is constant for the graph. Not all graphs may be so addressed. A weak characterization of addressable graphs in terms of the eigenvalues of a certain matrix associated with the graph is given. It is shown that any addressable bipartite graph may always be addressed with
d = 1
. For nonbipartite addressable graphs,
d
must be even, and it is shown that there exist graphs requiring an arbitrarily large
d
for addressing. An addressing algorithm is given which is guaranteed to address any addressable graph.