Boštjan Kuzman (Author), Aleksander Malnič (Author), Primož Potočnik (Author)

Abstract

V članku s pomočjo metode dviga avtomorfizmov v kontekstu elementarno-abelskih krovnih projekcij dopolnimo in posplošimo rezultate o štirivalentnih simetričnih grafih, ki sta jih obravnavala A. Gardiner in C. E. Praeger [Eur. J. Comb. 15, No. 4, 375--381 (1994)]. Vozliščno- in povezavno-tranzitivne grafe, katerih kvocient vzdolž normalne ▫$p$▫-elementarno abelske grupe avtomorfizmov za liho praštevilo ▫$p$▫ je cikel, so opisani s pomočjo cikličnih in negacikličnih kod. Natančneje, simetrijske lastnosti takšnih grafov so izpeljane iz določenih lastnosti polinomskih generatorjev cikličnih in negacikličnih kod, to je, iz deliteljev ▫$x^n\pm 1\in \mathbb{Z}_p [x]$▫. Ugotovitve uporabimo za kratek in poenoten opis tako razrešenih kot nerazrešenih primerov, ki sta jih obravnavala Gardiner in Praeger.

Keywords

tetravalent graphs;symmetric graphs;regular covers;cyclic codes;reflexible polynomials;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UL FMF - Faculty of Mathematics and Physics
UDC: 519.17
COBISS: 1540135620 Link will open in a new window
ISSN: 0095-8956
Views: 252
Downloads: 80
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Other data

Secondary language: English
Secondary title: Tetravalentni vozliščno- in povezavno- tranzitivni grafi nad podvojenimi cikli
Secondary abstract: In order to complete (and generalize) results of A. Gardiner and C. E. Praeger [Eur. J. Comb. 15, No. 4, 375--381 (1994)] on 4-valent symmetric graphs we apply the method of lifting automorphisms in the context of elementary-abelian covering projections. In particular, the vertex- and edge-transitive graphs whose quotient by a normal ▫$p$▫-elementary abelian group of automorphisms, for ▫$p$▫ an odd prime, is a cycle, are described in terms of cyclic and negacyclic codes. Specifically, the symmetry properties of such graphs are derived from certain properties of the generating polynomials of cyclic and negacyclic codes, that is, from divisors of ▫$x^n \pm 1 \in \mathbb{Z}_p [x]$▫. As an application, a short and unified description of resolved and unresolved cases of Gardiner and Praeger are given.
Type (COBISS): Article
Pages: str. 109-137
Issue: ǂVol. ǂ131
Chronology: July 2018
DOI: 10.1016/j.jctb.2018.01.007
ID: 10908756