Randić energy and Randić eigenvalues

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randić 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 Randić 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 Randić energy of its subdivided graph $S(G)$. We also give a necessary and sufficient condition for a graph to have exactly $k$ and distinct Randić eigenvalues.

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