TY - JOUR

T1 - Two Moore’s Theorems for Graphs

AU - Mednykh, Alexander

AU - Mednykh, Ilya

N1 - Publisher Copyright: © 2020. All Rights Reserved.

PY - 2020

Y1 - 2020

N2 - Let X be a finite connected graph, possibly with loops and multiple edges. An automorphism group of X acts purely harmonically if it acts freely on the set of directed edges of X and has no invert-ible edges. Define a genus g of the graph X to be the rank of the first homology group. A finite group acting purely harmonically on a graph of genus g is a natural discrete analogue of a finite group of automor-phisms acting on a Riemann surface of genus g. In the present paper, we investigate cyclic group (Formula presented)nacting purely harmonically on a graph X of genus g with fixed points. Given subgroup (Formula presented)d< (Formula presented)n, we find the signature of orbifold X/(Formula presented)dthrough the signature of orbifold X/(Formula presented)n. As a result, we obtain formulas for the number of fixed points for generators of group (Formula presented)dand for genus of orbifold X/(Formula presented)d. For Riemann surfaces, similar results were obtained earlier by M. J. Moore.

AB - Let X be a finite connected graph, possibly with loops and multiple edges. An automorphism group of X acts purely harmonically if it acts freely on the set of directed edges of X and has no invert-ible edges. Define a genus g of the graph X to be the rank of the first homology group. A finite group acting purely harmonically on a graph of genus g is a natural discrete analogue of a finite group of automor-phisms acting on a Riemann surface of genus g. In the present paper, we investigate cyclic group (Formula presented)nacting purely harmonically on a graph X of genus g with fixed points. Given subgroup (Formula presented)d< (Formula presented)n, we find the signature of orbifold X/(Formula presented)dthrough the signature of orbifold X/(Formula presented)n. As a result, we obtain formulas for the number of fixed points for generators of group (Formula presented)dand for genus of orbifold X/(Formula presented)d. For Riemann surfaces, similar results were obtained earlier by M. J. Moore.

KW - automorphism group

KW - graph

KW - harmonic action

KW - orbifold

UR - http://www.scopus.com/inward/record.url?scp=85108854523&partnerID=8YFLogxK

U2 - 10.13137/2464-8728/30918

DO - 10.13137/2464-8728/30918

M3 - Article

AN - SCOPUS:85108854523

VL - 52

SP - 469

EP - 476

JO - Rendiconti dell'Istituto di Matematica dell'Universita di Trieste

JF - Rendiconti dell'Istituto di Matematica dell'Universita di Trieste

SN - 0049-4704

ER -