A quadratic programming approach to the Randić index

Let G(k, n) be the set of connected graphs without multiple edges or loops which have n vertices and the minimum degree of vertices is k. The Randić index χ = χ(G) of a graph G   is defined by χ(G)=∑(uv)(δuδv)-1/2χ(G)=∑(uv)(δuδv)-1/2, where δu is the degree of vertex u and the summation extends over all edges (uv) of G. Caporossi et al. [G. Caporossi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs IV: Chemical trees with extremal connectivity index, Computers and Chemistry 23 (1999) 469–477] proposed the use of linear programming as one of the tools for finding the extremal graphs. In this paper we introduce a new approach based on quadratic programming for finding the extremal graphs in G(k, n) for this index. We found the extremal graphs or gave good bounds for this index when the number nk of vertices of degree k is between n − k and n. We also tried to find the graphs for which the Randić index attained its minimum value with given k (k ⩽ n/2) and n. We have solved this problem partially, that is, we have showed that the extremal graphs must have the number nk of vertices of degree k less or equal n − k and the number of vertices of degree n − 1 less or equal k.