Skew Randi\'c Matrix and Skew Randi\'c Energy

Abstract

Let $G$ be a simple graph with an orientation $\sigma$, which assigns to each edge a direction so that $G^\sigma$ becomes a directed graph. $G$ is said to be the underlying graph of the directed graph $G^\sigma$. In this paper, we define a weighted skew adjacency matrix with Rand\'c weight, the skew Randi\'c matrix ${\bf R_S}(G^\sigma)$, of $G^\sigma$ as the real skew symmetric matrix $[(r_s)_{ij}]$ where $(r_s)_{ij} = (d_id_j)^{-\frac{1}{2}}$ and $(r_s)_{ji} = -(d_id_j)^{-\frac{1}{2}}$ if $v_i \rightarrow v_j$ is an arc of $G^\sigma$, otherwise $(r_s)_{ij} = (r_s)_{ji} = 0$. We derive some properties of the skew Randi\'c energy of an oriented graph. Most properties are similar to those for the skew energy of oriented graphs. But, surprisingly, the extremal oriented graphs with maximum or minimum skew Randi\'c energy are completely different.