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The Randić index of a graph G, denoted by R(G), is defined as the sum of 1/d(u)d(v) over all edges uv of G, where d(u) denotes the degree of a vertex u in G. Caporossi and Hansen proposed a conjecture on the relation between the Randić index R(G) and the chromatic number χ(G) of a graph G: for any connected graph G of order n≥2, R(G)≥χ(G)−22+1n−1(χ(G)−1+n−χ(G)), and furthermore the bound is sharp...
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