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

TL;DR

This paper characterizes the eigenvalues of the Randić matrix for caterpillar graphs and identifies extremal structures that maximize or minimize Randić energy for specific cases.

Contribution

It provides necessary and sufficient conditions for eigenvalues of the Randić matrix of caterpillars and determines extremal caterpillars for Randić energy in several cases.

Findings

Characterization of eigenvalues for caterpillar graphs.

Identification of extremal caterpillars for Randić energy when r=2 and r=3.

Analysis of extremal structures within specific caterpillar families.

Abstract

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.