On the harmonic index and the chromatic number of a graph

The harmonic index H(G)H(G) of a graph GG is defined as the sum of the weights 2d(u)+d(v) of all edges uvuv of GG, where d(u)d(u) denotes the degree of a vertex uu in GG. The chromatic number χ(G)χ(G) of GG is the smallest number of colors needed to color all vertices of GG in such a way that no pair of adjacent vertices get the same color. The main result in this paper is χ(G)≤2H(G)χ(G)≤2H(G) proved by using the effect of removal of a minimum degree vertex on the harmonic index. It strengthens a result relating the Randić index and the chromatic number obtained by the system AutoGraphiX and proved by Hansen and Vukicević.