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Harmonic morphisms and hyperelliptic graphs. (English) Zbl 1178.05031
In this paper, authors study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. The famous Riemann-Hurwitz formula is a well known tool which is used in discrete group theory to calculate the genus of the subgroup using all data belonging to the group and subgroup. Here they reformulate a graph-theoretic analogue of the Riemann-Hurwitz formula. They also discuss several equivalent formulations of the notion of a hyperelliptic graph. They classify all hyperelliptic graphs having no Weierstarass points by using the Riemann-Hurwitz formula.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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