Total Chromatic Number and Some Topological Indices

The total chromatic number χ″(G)χ″(G) of G is the smallest number of colors needed to color all elements of G   in such a way that no adjacent or incident elements get the same color. The harmonic index H(G)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)d(u) denotes the degree of the vertex u in G. In this paper, we show a relation between the total chromatic number and the harmonic index. Also, we give relations between total chromatic number and some topological indices of a graph.