Duality Properties of Strong Isoperimetric Inequalities on a Planar Graph and Combinatorial Curvatures

Title
Duality Properties of Strong Isoperimetric Inequalities on a Planar Graph and Combinatorial Curvatures
Author
오병근
Keywords
Isoperimetric inequality; Planar graph; Combinatorial curvature; Gromov hyperbolicity
Issue Date
2014-06
Publisher
Springer Science + Business Media
Citation
DISCRETE & COMPUTATIONAL GEOMETRY; JUN 2014, 51 4, p859-p884, 26p.
Abstract
This paper is about hyperbolic properties on planar graphs. First, we study the relations among various kinds of strong isoperimetric inequalities on planar graphs and their duals. In particular, we show that a planar graph satisfies a strong isoperimetric inequality if and only if its dual has the same property, if the graph satisfies some minor regularity conditions and we choose an appropriate notion of strong isoperimetric inequalities. Second, we consider planar graphs where negative combinatorial curvatures dominate, and use the outcomes of the first part to strengthen the results of Higuchi, A >> uk, and, especially, Woess. Finally, we study the relations between Gromov hyperbolicity and strong isoperimetric inequalities on planar graphs, and give a proof that a planar graph satisfying a proper kind of a strong isoperimetric inequality must be Gromov hyperbolic if face degrees of the graph are bounded. We also provide some examples to support our results.
URI
http://link.springer.com/article/10.1007%2Fs00454-014-9592-7http://hdl.handle.net/20.500.11754/56497
ISSN
0179-5376
DOI
10.1007/s00454-014-9592-7
Appears in Collections:
COLLEGE OF EDUCATION[S](사범대학) > MATHEMATICS EDUCATION(수학교육과) > Articles
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