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  • A classification of the graphical ▫$m$▫-semiregular representation of finite groups
    Du, Jia-Li ; Feng, Yan-Quan ; Spiga, Pablo
    In this paper we extend the classical notion of digraphical and graphical regular representation of a group and we classify, by means of an explicit description, the finite groups satisfying this ... generalization. A graph or digraph is called regular if each vertex has the same valency, or, the same out-valency and the same in-valency, respectively. A (di)graphical ▫$m$▫-semiregular representation (respectively, ▫$\mathrm{G} m \mathrm{SR}$▫ and ▫$\mathrm{D} m \mathrm{SR}$▫, for short) of a group ▫$G$▫ is a regular (di)graph whose automorphism group is isomorphic to ▫$G$▫ and acts semiregularly on the vertex set with ▫$m$▫ orbits. When ▫$m = 1$▫, this definition agrees with the classical notion of GRR and DRR. Finite groups admitting a D1SR were classified by L. Babai [Period. Math. Hung. 11, 257-270 (1980)], and the analogue classification of finite groups admitting a G1SR was completed by C. D. Godsil [in: Algebraic methods in graph theory. Vol. I, II. Conference held in Szeged, August 24-31, 1978. Amsterdam-Oxford-New York: North-Holland Publishing Company. 221-239 (1981)]. Pivoting on these two results in this paper we classify finite groups admitting a ▫$\mathrm{G} m \mathrm{SR}$▫ or a ▫$\mathrm{D} m \mathrm{SR} $▫, for arbitrary positive integers ▫$m$▫. For instance, we prove that every non-identity finite group admits a ▫$\mathrm{G} m \mathrm{SR} $▫, for every ▫$m \geq 5$▫.
    Vir: Journal of combinatorial theory. Series A. - ISSN 0097-3165 (Vol. 171, Apr. 2020, art. 105174 (35 str.))
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2020
    Jezik - angleški
    COBISS.SI-ID - 27469827

    Povezava(-e):

    https://doi.org/10.1016/j.jcta.2019.105174
    DOI

vir: Journal of combinatorial theory. Series A. - ISSN 0097-3165 (Vol. 171, Apr. 2020, art. 105174 (35 str.))

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