- - - School of Mathematical Science, Shahrood University of Technology, Shahrood, Iran.
- - - School of Mathematical Science, Shahrood University of Technology, Shahrood, Iran.

‎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 ‎\sum_c D(G)=\sum |c(a)-c(b)| and \sum_s S(G)=\sum (c(a)+c(b))‎, ‎where the summations are taken over all edges ab\in E(G)‎. ‎The edge-difference chromatic sum‎, ‎denoted by \sum D(G)‎, ‎and the edge-sum chromatic sum‎, ‎denoted by \sum S(G)‎, ‎are respectively the minimum possible values‎ ‎of \sum_c D(G) and \sum_c S(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.

‎edge-difference chromatic sum‎, ‎edge-sum chromatic sum‎, ‎graph homomorphism‎, ‎Kneser graph‎, ‎fractional chromatic number