Skew Randic matrix and skew Randic energy

Let G G be a simple graph with an orientation σ σ ‎, ‎which ‎assigns to each edge a direction so that G σ Gσ becomes a‎ ‎directed graph‎. ‎G G is said to be the underlying graph of the‎ ‎directed graph G σ Gσ ‎. ‎In this paper‎, ‎we define a weighted skew‎ ‎adjacency matrix with Randc weight‎, ‎the skew Randic matrix ‎‎R S (G σ) ‎‎RS(Gσ) ‎, ‎of G σ Gσ as the real skew symmetric matrix‎ ‎[(r s) ij] [(rs)ij] where (r s) ij =(d i d j) −12 (rs)ij=(didj)−12 and‎ ‎(r s) ji =‎−‎(d i d j) −12 (rs)ji=‎−‎(didj)−12 if v i →v j vi→vj is‎ ‎an arc of G σ Gσ ‎, ‎otherwise (r s) ij =(r s) ji =0 (rs)ij=(rs)ji=0 ‎. ‎We‎ ‎derive some properties of the skew Randic 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 Randic energy are completely‎ ‎different‎, ‎no longer being some kinds of oriented regular graphs‎.