MONOTONICITY OF RESISTANCE DISTANCE IN LINEAR 2-TREES

Recall that a linear 2-tree, sometimes called a 2-path, is a 2-tree with exactly two vertices of degree two. In this article we will address two open questions regarding resistance distance in linear 2-trees. The first question is: given an arbitrary linear 2-tree does resistance distance between two vertices u,v increase as |v−u| increases? We answer this question in the affirmative. As a corollary to this result, we show that the maximal resistance distance in a linear 2-tree occurs between the vertices of degree 2 (the extremal vertices). The second question concerns the optimal location of bends in a linear 2-tree. We show that for a linear 2-tree with a single bend, the location of the bend that minimizes the maximal resistance distance (i.e., the resistance distance between the degree 2-vertices) is as close as possible to a degree 2 vertex. We show empirically and provide a conjecture that for a linear 2-tree with an arbitrary number of bends the configuration that will result in the smallest maximal resistance distance is to place the bends consecutively and as close as possible to one of the degree two vertices.