A proof of a conjecture on the Randić index of graphs with given girth

The Randić index R(G)R(G) of a graph GG is defined by R(G)=∑uv1d(u)d(v), where d(u) is the degree of a vertex uu in GG and the summation extends over all edges uvuv of GG. Aouchiche, Hansen and Zheng proposed the following conjecture: For any connected graph on n≥3n≥3 vertices with Randić index RR and girth gg, R+g≥n−3+2n−1+72andR⋅g≥3n−9+32n−1+32 with equalities if and only if G=Sn+. This paper is devoted to giving a confirmative proof to this conjecture.