On Dense Graphs Having Minimum Randic Index

In this paper all graphs G of order n and minimum degree delta(G) = k having minimum Randic index R(G) are determined for k >= left perpendicularn/2right perpendicular. Each extremal graph is the join between a regular graph of order n - s and a complete graph of order s (where s is an element of{n/2, (n+2)/2, (n-2)/2} for n even and s is an element of {(n+1)/2, (n-1)/2} for n odd). This yields an alternative proof in the case of dense graphs to that proposed by Li, Liu and Liu [5] who very recently solved a long-standing conjecture on Randic index. Also, the minimum value of this index in the class of graphs of order n and delta(G) = k is determined for k >= (n-1)/2.