Genus Bounds for Harmonic Group Actions on Finite Graphs

Abstract

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).

Critical groups of graphs with dihedral actions II

“…In fact, an explicit calculation shows that Jac(G) ∼ = Z/3Z ⊕ Z/2Z ⊕ (Z/8Z) 2 . We note that this graph is also the circulant graph C 1,2 6 and as such it also admits an action by the group D 3 ; one can derive the same result by decomposing the graph according to this action and Theorem 6.3.…”

On Wiman’s theorem for graphs

“…Then the upper bound on the order of a group acting harmonically on a graph of genus g > 1 is 6(g − 1). This result was obtained by S.Corry [5].The aim of the present paper is to find a discrete version of the Wiman theorem.Let Z N be a cyclic group acting freely on the set of directed edges of a graph X of genus g ≥ 2. We prove that N ≤ 2g + 2.…”

“…Around the same time, Wiman [15] characterized the curves w 2 = z 2g+1 − 1 and w 2 = z(z 2g − 1), g > 1, as the unique curves of genus g admitting cyclic automorphism groups of the largest and the second largest possible order (4g + 2 and 4g, respectively). The modern proof of these and similar results is contained in the paper by K. Nakagawa [13].Over the last decade, counterparts of many theorems from the classical theory of Riemann surfaces were derived in the discrete case [1,3,5,12]. In these theorems, the finite connected graphs play the role of algebraic curves and the conformal automorphisms are replaced by harmonic ones.…”

“…Over the last decade, counterparts of many theorems from the classical theory of Riemann surfaces were derived in the discrete case [1,3,5,12]. In these theorems, the finite connected graphs play the role of algebraic curves and the conformal automorphisms are replaced by harmonic ones.…”

Isospectral genus two graphs are isomorphic

“…Under these assumptions, the theory of Jacobi manifolds is constructed and analogues of the Riemann-Hurwitz and Riemann-Roch theorems were proved. Counterparts of many other theorems from the classical theory of Riemann surfaces were derived in the discrete case ( [9,10,16]). …”