  Acharya introduced the notion of set-valuations of graphs as a set analogue
of the number valuations of graphs. Also we have the notion of set-indexers,
integer additive set-indexers and k-uniform integer additive set-indexers. In
this paper, we initiate a study of the graphs which admit k-uniform integer
additive set-indexers as a generalisation to the earlier studies of 1-uniform
and 2-uniform integer additive set-indexers. We introduce the notion of weakly
uniform integer additive set-indexers and arbitrarily uniform integer additive
set-indexers based on the cardinality of the labeling sets and provide some
useful results on these types of set-indexers.
