Randic energy of specific graphs

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 $G$.