Randic energy of specific graphs

TL;DR

This paper computes the Randić characteristic polynomial and Randić energy for specific graphs, providing insights into the spectral properties of the Randić matrix associated with these graphs.

Contribution

It introduces explicit calculations of the Randić characteristic polynomial and energy for particular classes of graphs, advancing spectral graph theory.

Findings

Derived formulas for Randić characteristic polynomial.

Calculated Randić energy for specific graph families.

Enhanced understanding of spectral properties of Randić matrices.

Abstract

Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,..., v_n\}$. The Randi\'{c} matrix of $G$, denoted by $R(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $(d_id_j)^{\frac{-1}{2}}$ if $v_i$ and $v_j$ are adjacent and $0$ for another cases. Let the eigenvalues of the Randi\'{c} matrix $R(G)$ be $\rho_1\geq \rho_2\geq ...\geq \rho_n$ which are the roots of the Randi\'c characteristic polynomial $\prod_{i=1}^n (\rho-\rho_i)$. The Randi\'{c} energy $RE$ of $G$ is the sum of absolute values of the eigenvalues of $R(G)$. In this paper we compute the Randi\'c characteristic polynomial and the Randi\'c energy for specific graphs $G$.