On the Edge-Difference and Edge-Sum Chromatic Sum of the Simple Graphs

For a coloring c of a graph G, the edge-difference coloring sum and edge-sum coloring sum with respect to the coloring c are respectively ∑cD(G)=∑|c(a)−c(b)| and ∑sS(G)=∑(c(a)+c(b)), where the summations are taken over all edges ab∈E(G). The edge-difference chromatic sum, denoted by ∑D(G), and the edge-sum chromatic sum, denoted by ∑S(G), are respectively the minimum possible values of ∑cD(G) and ∑cS(G), where the minimums are taken over all proper coloring of c. In this work, we study the edge-difference chromatic sum and the edge-sum chromatic sum of graphs. In this regard, we present some necessary conditions for the existence of homomorphism between two graphs. Moreover, some upper and lower bounds for these parameters in terms of the fractional chromatic number are introduced as well.