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The present paper is devoted to the further development of the discrete theory of Riemann surfaces, which was started in the papers by M. Baker and S. Norine and their followers at the beginning of the century. This theory considers finite graphs as analogs of Riemann surfaces and branched coverings of graphs as holomorphic mappings. The genus of a graph is defined as the rank of its fundamental group. The main object of investigation in the paper is automorphism groups of a graph acting freely on the set of half-edges of the graph. These groups are discrete analogs of groups of conformal automorphisms of a Riemann surface. The celebrated Hurwitz theorem (1893) states that the order of the group of conformal automorphisms of a compact Riemann surface of genus g > 1 does not exceed 84(g — 1). Later, K. Oikawa and T. Arakawa refined this bound in the case of groups that fix several finite sets of prescribed cardinalities. This paper provides proofs of discrete versions of the mentioned theorems. In addition, a discrete analog of the E. Bujalance and G. Gromadzki theorem improving one of Arakawa's results is obtained. [ABSTRACT FROM AUTHOR]

The isomorphism classes of hypermaps of a given genus g ≤ 6 and a given number d of darts are enumerated. The hypermaps of a given genus g are distinguished up to orientation preserving isomorphisms. The obtained results depend on recent progress in counting rooted hypermaps, in particular, by P. Zograf, M. Kazarian, A. Giorgetti, and T. Walsh. These results can be interpreted as an enumeration of conjugacy classes of subgroups in a free Fuchsian group of rank two with a genus restriction. Bibliography: 36 titles. [ABSTRACT FROM AUTHOR]

Several versions of the Riemann-Hurwitz theorem for branched coverings of graphs are presented. A larger class of graphs which may have not only multiple edges but also loops and darts is considered. This makes it possible to render the class of graphs closed with respect to morphisms arising as quotient maps for actions of finite groups. [ABSTRACT FROM AUTHOR]

The article discusses the Hurwitz' Theorem or the upper bound for groups acting on a Riemann surface. Topics include role of holomorphic mappings and Riemann surfaces, finite multigraphs and their automorphism groups, and rank of finite multigraph's fundamental groups to define the genus of graphs.