Orthogonal covers by multiplication graphs

Hans-Dietrich Gronau; Markus Schmidmeier

doi:10.18453/rosdok_id00005832

zum übergeordneten Dokument

Hans-Dietrich Gronau Markus Schmidmeier

Orthogonal covers by multiplication graphs

University of Rostock, Institute of Mathematics, 2007

https://doi.org/10.18453/rosdok_id00005832

Abstract: Let K be the complete oriented graph on the finite set of vertices A. A family G = {Ga : a ∈ A} of spanning subgraphs of K is an orthogonal cover provided every arrow of K occurs in exactly one Ga and for every two elements a, b ∈ A, the graphs Ga and Gopb have exactly one arrow in common. Gronau, Grüttmüller, Hartmann, Leck and Leck (2002) have observed that if A has the structure of a finite ring and if f ∈ A is such that both f + 1 and f − 1 are units, then the family, obtained by taking for G0 the multiplication graph of f and for Ga the rotation of G0 by a, defines an orthogonal cover on K. In this manuscript we assume that A is a finite abelian group and proceed to (i) generalize this construction to arbitrary endomorphisms of the underlying group and describe the possible graphs, (ii) introduce a duality on the set of orthogonal covers and (iii) give detailed descriptions of the covers in the case where A is cyclic or elementary abelian.

Workingpaper Freier Zugang

Titel:

Orthogonal covers by multiplication graphs

Schriftenreihe:

Preprints aus dem Institut für Mathematik, 07/1

Beteiligte Personen:

Hans-Dietrich Gronau[VerfasserIn]
	138612498
Markus Schmidmeier[VerfasserIn]
	0000-0003-3365-6666
	173040292

Zusammenfassung:

Let K be the complete oriented graph on the finite set of vertices A. A family G = {Ga : a ∈ A} of spanning subgraphs of K is an orthogonal cover provided every arrow of K occurs in exactly one Ga and for every two elements a, b ∈ A, the graphs Ga and Gopb have exactly one arrow in common. Gronau, Grüttmüller, Hartmann, Leck and Leck (2002) have observed that if A has the structure of a finite ring and if f ∈ A is such that both f + 1 and f − 1 are units, then the family, obtained by taking for G0 the multiplication graph of f and for Ga the rotation of G0 by a, defines an orthogonal cover on K. In this manuscript we assume that A is a finite abelian group and proceed to (i) generalize this construction to arbitrary endomorphisms of the underlying group and describe the possible graphs, (ii) introduce a duality on the set of orthogonal covers and (iii) give detailed descriptions of the covers in the case where A is cyclic or elementary abelian. [Englisch]

Dokumenttyp:

Workingpaper

Einrichtung:

Mathematisch-Naturwissenschaftliche Fakultät

Sprache:

Englisch

Sachgruppe der DNB:

510 Mathematik

Umfang:

1 Online-Ressource (15 Seiten)

Veröffentlichung /
Entstehung:

Rostock: University of Rostock, Institute of Mathematics