Abstract
A caterpillar graph \(T(p_1,\ldots, p_r)\) of order \(n = r + \sum^r_{i=1} p_i , r \geq 2\), is a tree such that removing all its pendent vertices gives rise to a path of order \(r\). In this paper we establish a necessary and sufficient condition for a real number to be an eigenvalue of the Randić matrix of \(T(p_1,\ldots, p_r)\). This result is applied to determine the extremal caterpillars for the Randić energy of \(T(p_1,\ldots, p_r)\) for cases \(r = 2\) (the double star) and \(r = 3\). We characterize the extremal caterpillars for \(r = 2\). Moreover, we study the family of caterpillars \(T(p, n − p − q − 3, q)\) of order \(n\), where \(q\) is a function of \(p\), and we characterize the extremal caterpillars for three cases: \(q = p, q = n − p − b − 3 \ \text{and} \ q = b\), for \(b \in \{1,\ldots, n − 6\}\) fixed. Some illustrative examples are included.