Randi\'c energy and Randi\'c eigenvalues

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi'c matrix $\mathbf{R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 / \sqrt{d_jd_j}$ifthevertices$v_i$and$v_j$areadjacentin$G$and$r_{ij}=0$otherwise\mathrm{.}ThenormalizedsignlessLaplacianmatrix$$\mathcal{Q}$$ is defined as $\mathcal{Q}=I+\mathbf{R}$, where $I$ is the identity matrix. The Randi'c energy is the sum of absolute values of the eigenvalues of $\mathbf{R}$. In this paper, we find a relation between the normalized signless Laplacian eigenvalues of $G$ and the Randi'c energy of its subdivided graph $S(G)$. We also give a necessary and sufficient condition for a graph to have exactly $k$ and distinct Randi'c eigenvalues.