Randic energy of specific graphs

Abstract

Let $G$ be a simple graph with vertex set $V(G)=\{v_1,v_2,...,v_n\}$. The Randić matrix of $G$, denoted by $R(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $(d_i{d}_j{)}^{\frac{-1}{2}}$ if $v_i$ and $v_j$ are adjacent and $0$ for another cases. Let the eigenvalues of the Randić matrix $R(G)$ be ${\rho }_1\ge {\rho }_2\ge ...\ge {\rho }_n$ which are the roots of the Randić characteristic polynomial ${\prod }_{i=1}^n(\rho -{\rho }_i)$. The Randić energy $RE$ of $G$ is the sum of absolute values of the eigenvalues of $R(G)$. In this paper we compute the Randić characteristic polynomial and the Randić energy for specific graphs $G$.

15 pages, 2 figures