Some New Bounds on Randić Energy

E. Zogić, B. Borovićanin

Let $G=(V,E)$ be a simple graph of order $n$ with vertex set $V=V(G)=\{v_1,v_2,\ldots,v_n\}$ and edge set $E=E(G)$. Let $d_i$ be the degree of the vertex $v_i$ in $G$ for $i=1,2,\ldots, n$. The Randić matrix ${\mathbf R}={\mathbf R}(G)=||R_{ij}||_{nxn}$ is defined by $$R_{ij}=eft\{\begin{array}{cl} \dfrac{1}{qrt{d_id_j}}, & \mbox{if the vertices}\hspace{0.1cm} v_i\hspace{0.1cm}\mbox{and}\hspace{0.1cm} v_j\hspace{0.1cm} \mbox{are adjacent}, 0, & \mbox{otherwise}. \end{array}\right.$$ The eigenvalues of matrix ${\mathbf R}$, denoted by $\rho_1,\rho_2,\ldots,\rho_n$, are called the Randić eigenvalues of graph $G$. The Randić energy of graph $G$, denoted by $RE$, is defined as $$RE=RE(G)=umimits_{i=1}^n|\rho_i|.$$ In this paper we establish some new upper and lower bounds on Randić energy.