Author:
Amini Omid,Manjunath Madhusudan
Abstract
Recently, Baker and Norine (Advances in Mathematics, 215(2): 766-788, 2007) found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph $G$. In this paper, we develop a general Riemann-Roch theory for sub-lattices of the root lattice $A_n$ analogue to the work of Baker and Norine, and establish connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In this way, we obtain a geometric proof of the Riemann-Roch theorem for graphs and generalise the result to other sub-lattices of $A_n$. In particular, we provide a new geometric approach for the study of the Laplacian of graphs. We also discuss some problems on classification of lattices with a Riemann-Roch formula as well as some related algorithmic issues.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
11 articles.
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