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Conformal Geometry and Dynamics

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Modulus on graphs as a generalization of standard graph theoretic quantities


Authors: Nathan Albin, Megan Brunner, Roberto Perez, Pietro Poggi-Corradini and Natalie Wiens
Journal: Conform. Geom. Dyn. 19 (2015), 298-317
MSC (2010): Primary 90C35
DOI: https://doi.org/10.1090/ecgd/287
Published electronically: December 4, 2015
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Abstract: This paper presents new results for the modulus of families of walks on a graph--a discrete analog of the modulus of curve families due to Beurling and Ahlfors. Particular attention is paid to the dependence of the modulus on its parameters. Modulus is shown to generalize (and interpolate among) three important quantities in graph theory: shortest path, effective resistance, and max-flow or min-cut.


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Nathan Albin
Affiliation: Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, Kansas 66506
Email: albin@math.ksu.edu; pietro@math.ksu.edu

Megan Brunner
Affiliation: Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, Kansas 66506

Roberto Perez
Affiliation: Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, Kansas 66506

Pietro Poggi-Corradini
Affiliation: Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, Kansas 66506

Natalie Wiens
Affiliation: Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, Kansas 66506

DOI: https://doi.org/10.1090/ecgd/287
Keywords: Modulus of families of walks, effective resistance, shortest path, max-flow, min-cut
Received by editor(s): June 1, 2015
Received by editor(s) in revised form: October 30, 2015
Published electronically: December 4, 2015
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. 126287 (Albin, Brunner, Perez, Wiens), through Kansas State University’s 2014 Summer Undergraduate Mathematics Research program, and under Grant Nos. 1201427 (Poggi-Corradini) and 1515810 (Albin)
Article copyright: © Copyright 2015 American Mathematical Society

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