How much does a Hamiltonian cycle weigh?

1. INTRODUCTION. A Hamiltonian cycle in a graph is a path in the graph which visits each vertex exactly once and returns to the starting vertex. Let í µí°¾ í µí±› be a weighted complete graph with í µí±› vertices. We define the weight of an edge as the square of the distance between two end points of the edge. The weight of a path í µí±ƒ is the sum of the weights of all edges in the path and denoted by í µí±¤(í µí±ƒ).