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

TL;DR

This paper introduces the skew Randić matrix for oriented graphs, explores its properties, and reveals that extremal graphs for maximum or minimum skew Randić energy differ significantly from those for skew energy.

Contribution

It defines the skew Randić matrix for oriented graphs and analyzes its properties, highlighting differences from the skew energy of oriented graphs.

Findings

Properties of skew Randić energy derived

Extremal graphs for skew Randić energy differ from skew energy

Surprising differences in extremal graph structures

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.