Generalised colouring sums of graphs

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 $ \varphi ^{\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 $ \chi $-chromatic sum, $ \chi ^+ $-chromatic sum, $ b^+ $-chromatic sum, $ \pi $-chromatic sum and $ \pi ^+ $-chromatic sum of graphs. We also discuss certain results on these parameters for a selection of standard graphs