Randi\'c energy and Randi\'c eigenvalues

TL;DR

This paper explores the relationship between Randić energy and normalized signless Laplacian eigenvalues in graphs, providing conditions for the number of distinct Randić eigenvalues and analyzing subdivided graphs.

Contribution

It establishes a link between signless Laplacian eigenvalues and Randić energy of subdivided graphs, and characterizes graphs with a specific number of distinct Randić eigenvalues.

Findings

Relation between eigenvalues of $\mathcal{Q}$ and Randić energy of $S(G)$

Necessary and sufficient conditions for graphs with exactly $k$ distinct Randić eigenvalues

Analysis of subdivided graphs' spectral properties

Abstract

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi\'c matrix ${\bf R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 / \sqrt{d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and $r_{ij}=0$ otherwise. The normalized signless Laplacian matrix $\mathcal{Q}$ is defined as $\mathcal{Q} =I+\bf{R}$, where $I$ is the identity matrix. The Randi\'c energy is the sum of absolute values of the eigenvalues of $\bf{R}$. In this paper, we find a relation between the normalized signless Laplacian eigenvalues of $G$ and the Randi\'c energy of its subdivided graph $S(G)$. We also give a necessary and sufficient condition for a graph to have exactly $k$ and distinct Randi\'c eigenvalues.