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

ýFor a coloring c of a graph G ý, ýthe edge-difference coloring sum and edge-sum coloring sum with respect to the coloring c are respectivelyý ý∑ c D(G)=∑|c(a)−c(b)| and ∑sS(G)=∑(c(a)(b))ý, ýwhere the summations are taken over all edges ab∈E(G)ý.
ýThe edge-difference chromatic sumý, ýdenoted by ∑D(G) ý, ýand the edge-sum chromatic sumý, ýdenoted by ∑S(G) ý, ýare respectively the minimum possible valuesý ýof ∑ c D(G) and ∑ c S(G)ý, ýwhere the minimums are taken over all proper coloring of c ý.ýIn this worký, ýwe study the edge-difference chromatic sum and the edge-sum chromatic sum of graphsý. ýIn this regardý,ýwe present some necessary conditions for the existence of homomorphism between two graphsý. ýMoreoverý, ýsome upper and lower bounds for these parameters in terms of the fractional chromatic number are introducedý ýas wellý.