Recursive Formulas for Beans Functions of Graphs

In this paper, we regard each edge of a connected graph G as a line segment having a unit length, and focus on not only the “vertices” but also any “point” lying along such a line segment. So we can define the distance between two points on G as the length of a shortest curve joining them along G . The beans function B G ( x ) of a connected graph G is defined as the maximum number of points on G such that any pair of points have distance at least x > 0. We shall show a recursive formula for B G ( x ) which enables us to determine the value of B G ( x ) for all x ≤ 1 by evaluating it only for 1 / 2 < x ≤ 1. As applications of this recursive formula, we shall propose an algorithm for computing B G ( x ) for a given value of x ≤ 1, and determine the beans functions of the complete graphs K n .