Skew Randić Matrix and Skew Randić 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 Randć weight, the skew Randić matrix ${\mathbf{R}}_{\mathbf{S}}(G^{\sigma })$, of $G^{\sigma }$ as the real skew symmetric matrix $[(r_s{)}_{ij}]$ where $(r_s{)}_{ij}=(d_i{d}_j{)}^{-\frac{1}{2}}$ and $(r_s{)}_{ji}=-(d_i{d}_j{)}^{-\frac{1}{2}}$ if $v_i\to v_j$ is an arc of $G^{\sigma }$, otherwise $(r_s{)}_{ij}=(r_s{)}_{ji}=0$. We derive some properties of the skew Randić 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 Randić energy are completely different.

16 pages