On the Randić index and girth of graphs

The Randić index R(G)R(G) of a graph GG is defined by R(G)=∑uv1d(u)d(v), where d(u)d(u) is the degree of a vertex uu in GG and the summation extends over all edges uvuv of GG. In this work, we give a sharp upper bound and a lower bound of the Randić index among connected nn-vertex graphs with girth g≥k(k≥3)g≥k(k≥3).