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.

The notion of the b-chromatic number of a graph attracted much research interests and recently a new concept, namely the b-chromatic sum of a graph, denoted by ${\mathit{\phi }}^{\prime }(G)$, has also been introduced. Motivated by the studies on b-chromatic sum of graphs, in this paper we introduce certain new parameters such as $\mathit{\chi }$-chromatic sum, ${\mathit{\chi }}^{+}$-chromatic sum, $b^{+}$-chromatic sum, $\mathit{\pi }$-chromatic sum and ${\mathit{\pi }}^{+}$-chromatic sum of graphs. We also discuss certain results on these parameters for a selection of standard graphs.