More on energy and Randic energy of specific graphs

TL;DR

This paper investigates the energy and Randić energy of specific graphs, providing calculations for certain cases and proposing a new conjecture on Randić energy properties.

Contribution

It introduces formulas for energy and Randić energy of particular graphs and presents a new conjecture on Randić energy behavior.

Findings

Computed energy and Randić energy for specific graphs

Proposed a new conjecture on Randić energy

Enhanced understanding of spectral properties of graphs

Abstract

Let $G$ be a simple graph of order $n$. The energy $E(G)$ of the graph $G$ is the sum of the absolute values of the eigenvalues of $G$. 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. 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 energy and Randi\'{c} energy for certain graphs. Also we propose a conjecture on Randi\'c energy.