Skew Randi'c matrix and skew Randi'c energy

Document Type : Research Paper

Authors

1 Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China

2 Center for Combinatorics, Nankai University, Tianjin 300071, China

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, ‎no longer being some kinds of oriented regular graphs‎.

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References

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History

  • Receive Date: 27 December 2014
  • Revise Date: 14 May 2015
  • Accept Date: 14 May 2015
  • Published Online: 01 March 2016

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