A Note on Chromatic Sum
  The chromatic sum $\Sigma(G)$ of a graph $G$ is the smallest sum of colors among of proper coloring with the natural number. In this paper, we introduce a necessary condition for the existence of graph homomorphisms. Also, we present $\Sigma(G)<\chi_f(G)|G|$ for every graph $G$. 
Meysam Alishahi and Ali Taherkhani