摘要翻译：
我们研究了图的调和态射作为Riemann曲面之间全纯映射的自然离散模拟。我们建立了经典Riemann-Hurwitz公式的图论模拟，研究了由调和态射引起的Jacobian型和调和1-型上的泛函映射，并给出了从Riemann曲面到射影空间的正则映射的离散模拟。我们还讨论了超椭圆图概念的几个等价公式，它们都是由经典黎曼曲面理论所推动的。作为我们结果的应用，我们证明了对于不是圈的2-边连通图G，在G上至多存在一个对合$\Iota$，其商$G/\Iota$是树。我们还证明了图G中生成树的个数是偶数的当且仅当G对由2条边连接的2个顶点组成的图B2允许一个非常调和态射。最后，利用Riemann-Hurwitz公式和我们关于超椭圆图的结果，对所有不含Weierstrass点的超椭圆图进行了分类。
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英文标题：
《Harmonic morphisms and hyperelliptic graphs》
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作者：
Matthew Baker and Serguei Norine
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Combinatorics        组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involution $\iota$ on G for which the quotient $G/\iota$ is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a non-constant harmonic morphism to the graph B_2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.
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PDF链接：
https://arxiv.org/pdf/0707.1309