Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi\'c matrix ${ \bf R}=(r_{ ij})$ of $G$ is defined by $r_{ ij} = 1 / \sqrt{ d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and $r_{ ij}=0$ otherwise. The normalized signless Laplacian matrix $\mathcal{ Q}$ is defined as $\mathcal{ Q} =I+\bf{ R}$, where $I$ is the identity matrix. The Randi\'c energy is the sum of absolute values of the eigenvalues of $\bf{ R}$. 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.
Categories: math.CO