On solving a non-convex quadratic programming problem involving resistance distances in graphs

Quadratic programming problems involving distance matrix (D) that arises in trees are considered in the literature by Dankelmann (Discrete Math 312:12-20, 2012), Bapat and Neogy (Ann Oper Res 243:365-373, 2016). In this paper, we consider the question of solving the quadratic programming problem of finding maximum of x(T)Rx subject to x being a nonnegative vector with sum 1 and show that for the class of simple graphs with resistance distance matrix (R) which are not necessarily a tree, this problem can be reformulated as a strictly convex quadratic programming problem. An application to symmetric bimatrix game is also presented.