Characterizing When Powers of a Caterpillar are Divisor Graphs

Abstract

In this paper, we prove that for any tree \(T\), \(T^2\) is a divisor graph if and only if \(T\) is a caterpillar and the diameter of \(T\) is less than six. For any caterpillar \(T\) and a positive integer \(k \geq 1\) with \(diam(T) \leq 2k\), we show that \(T^k\) is a divisor graph. Moreover, for a caterpillar \(T\) and \(k \geq 3\) with \(diam(T) = 2k\) or \(diam(T) = 2k + 1\), we show that \(T^k\) is a divisor graph if and only if the centers of \(T\) have degree two.