On the Randi\'{c} energy of caterpillar graphs

A caterpillar graph $T(p_1, \ldots, p_r)$ of order $n= r+\sum_{i=1}^r p_i$, $r\geq 2$, 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 Randi\'c matrix of $T(p_1, \ldots, p_r)$. This result is applied to determine the extremal caterpillars for the Randi\'c 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 $T\big(p,n-p-q-3,q\big)$ of order $n$, where $q$ is a function of $p$, and we characterize the extremal caterpillars for three cases: $q=p$, $q=n-p-b-3$ and $q=b$, for $b\in \{1,\ldots,n-6\}$ fixed. Some illustrative examples are included.