Skew Randi'c matrix and skew Randi'c energy

• Main Authors: Ran Gu, Fei Huang, Xueliang Li
• Format: Article
• Language: English
• Published: University of Isfahan 2016-03-01
• Series: Transactions on Combinatorics
• Subjects: oriented graph; skew Randi'c matrix; skew Randi'c energy
• Online Access: http://www.combinatorics.ir/article_9513_5dd2d75be3009b50ce663bc68f39cb1e.pdf

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, ‎no longer being some kinds of oriented regular graphs‎.