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Symmetry 2011, 3(3), 564-573; doi:10.3390/sym3030564
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
Received: 2 March 2011; in revised form: 21 July 2011 / Accepted: 28 July 2011 / Published: 15 August 2011
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Abstract: 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.
Keywords: graph theory; digraph symmetries; semigroup; Green’s relations; structural invariance
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MDPI and ACS Style
Parks, A.D. Green’s Symmetries in Finite Digraphs. Symmetry 2011, 3, 564-573.AMA Style
Parks AD. Green’s Symmetries in Finite Digraphs. Symmetry. 2011; 3(3):564-573.Chicago/Turabian Style
Parks, Allen D. 2011. "Green’s Symmetries in Finite Digraphs." Symmetry 3, no. 3: 564-573.