Abstract

The harmonic index H(G) of a graph G is defined as the sum of the weights 2d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. The chromatic number χ(G) of G is the smallest number of colors needed to color all vertices of G 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) 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.

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