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Abstract. This paper develops graph analogues of the genus bounds for the maximal size of an automorphism group of a compact Riemann surface of genus g ≥ 2. Inspired by the work of M. Baker and S. Norine on harmonic morphisms between finite graphs, we motivate and define the notion of a harmonic group action. Denoting by M (g) the maximal size of such a harmonic group action on a graph of genus g ≥ 2, we prove that 4(g − 1) ≤ M (g) ≤ 6(g − 1), and these bounds are sharp in the sense that both are attained for infinitely many values of g. Moreover, we show that the values 4(g − 1) and 6(g − 1) are the only values taken by the function M (g).

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag(d) − A)r = 0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag(d)−A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients 2n−1 n−1 , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

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This paper studies groups of maximal size acting harmonically on a finite graph. Our main result states that these maximal graph groups are exactly the finite quotients of the modular group $\Gamma=\left$ of size at least 6. This characterization may be viewed as a discrete analogue of the description of Hurwitz groups as finite quotients of the $(2,3,7)$-triangle group in the context of holomorphic group actions on Riemann surfaces. In fact, as an immediate consequence of our result, every Hurwitz group is a maximal graph group, and the final section of the paper establishes a direct connection between maximal graphs and Hurwitz surfaces via the theory of combinatorial maps.Comment: 17 pages, 6 figures. Final section rewritten to emphasize connection to combinatorial map