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Isoperimetric Constants of Infinite Plane Graphs |
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| Authors: | Serge Lawrencenko Michael D Plummer Xiaoya Zha | |
| Institution: | (1) Department of Mathematics and Statistics, Rochester Institute of Technology, Rochester, NY 14623, USA , US;(2) Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA michael.d.plummer@vanderbilt.edu , US;(3) Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37132, USA xzha@.mtsu.edu, US | |
| Abstract: |
Abstract. Let G be an infinite locally finite plane graph with one end and let H be a finite plane subgraph of G . Denote by a(H) the number of finite faces of H and by l(H) the number of the edges of H that are on the boundary of the infinite face or a finite face not in H . Define the isoperimetric constant h (G) to be inf
H
l(H) / a(H) and define the isoperimetric constant h (δ) to be inf
G
h (G) where the infimum is taken over all infinite locally finite plane graphs G having minimum degree δ and exactly one end. We establish the following bounds on h (δ) for δ ≥ 7 :
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