  An integer additive set-indexer is defined as an injective function
$f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G)
\rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also
injective. An integer additive set-indexer is said to be $k$-uniform if
$|g_f(e)| = k$ for all $e\in E(G)$. An integer additive set-indexer $f$ is said
to be a weak integer additive set-indexer if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for
all $u,v\in V(G)$. In this paper, we study the characteristics of certain
graphs and graph classes which admit weak integer additive set-indexers.
