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Green’s Symmetries in Finite Digraphs

Electromagnetic and Sensor Systems Department, 18444 Frontage Road Suite 327, Naval Surface Warfare Center Dahlgren Division, Dahlgren, VA 22448-5161, USA
Symmetry 2011, 3(3), 564-573;
Received: 2 March 2011 / Revised: 21 July 2011 / Accepted: 28 July 2011 / Published: 15 August 2011
PDF [331 KB, uploaded 15 August 2011]


The semigroup DV of digraphs on a set V of n labeled vertices is defined. It is shown that DV is faithfully represented by the semigroup Bn of n ´ n Boolean matrices and that the Green’s L, R, H, and D equivalence classifications of digraphs in DV follow directly from the Green’s classifications already established for Bn. The new results found from this are: (i) L, R, and H equivalent digraphs contain sets of vertices with identical neighborhoods which remain invariant under certain one-sided semigroup multiplications that transform one digraph into another within the same equivalence class, i.e., these digraphs exhibit Green’s isoneighborhood symmetries; and (ii) D equivalent digraphs are characterized by isomorphic inclusion lattices that are generated by their out-neighborhoods and which are preserved under certain two-sided semigroup multiplications that transform digraphs within the same D equivalence class, i.e., these digraphs are characterized by Green’s isolattice symmetries. As a simple illustrative example, the Green’s classification of all digraphs on two vertices is presented and the associated Green’s symmetries are identified. View Full-Text
Keywords: graph theory; digraph symmetries; semigroup; Green’s relations; structural invariance graph theory; digraph symmetries; semigroup; Green’s relations; structural invariance

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Parks, A.D. Green’s Symmetries in Finite Digraphs. Symmetry 2011, 3, 564-573.

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