FACILITY LOCATION NETWORK IN TERMS OF DISTANCE IN GRAPHS

Let G be an (n, m) graph. Average distance defined as the sum of the distances between all unordered pairs of vertices by n 2 In this paper, we consider famous problem in facility location network, where, its aim is to partition the nodes of a network considered into a set of facility nodes and a set of customer nodes, such that the average distance between a facility and a customer is minimized. Let P = {V 1 , . . . , V k } be a partition of V (G) and d(V i ) shows the sum of distances between vertices of V i , then, the partition distance of G is defined as the summation of d(V i )s for all 1 ≤ i ≤ k. This concept generalizes several metric concepts and is dual to the concept of the colored distance due to Dankelmann, Goddard, and Slater. We prove our main results which assert that the partition distance of a graph can be obtained from certain smaller weighted graphs. Also show that numerous earlier results follow directly from the obtained theorems. Different general upper bounds on the partition distance of trees are proved. Trees that attain the respective bounds are also characterized. We conclude with some remarks on the relation between the partition distance and the colored distance and on the concept of the k-diameter from Goddard et al. 2005.