University of Isfahan Transactions on Combinatorics 2251-8657 5 1 2016 03 01 Skew Randi'c matrix and skew Randi'c energy 1 14 9513 10.22108/toc.2016.9513 EN Ran Gu Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China Fei Huang Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China Xueliang Li Center for Combinatorics, Nankai University, Tianjin 300071, China Journal Article 2014 12 27 ‎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. https://toc.ui.ac.ir/article_9513_5dd2d75be3009b50ce663bc68f39cb1e.pdf