Vol. 3, No. 1, 2010

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ISSN: 1944-4184 (e-only)
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Roundness properties of graphs

Matthew Horak, Eric LaRose, Jessica Moore, Michael Rooney and Hannah Rosenthal

Vol. 3 (2010), No. 1, 67–91
Abstract

The notion of the roundness of a metric space was introduced by Per Enflo as a tool to study geometric properties of Banach spaces. Recently, roundness and generalized roundness have been used in the context of group theory to investigate relationships between the geometry of a Cayley graph of a group and the algebraic properties of the group. In this paper, we study roundness properties of connected graphs in general. We explicitly calculate the roundness of members of two classes of graphs and we give results of computer calculations of the roundness of all connected graphs on 7, 8 and 9 vertices. We also show that no connected graph can have roundness between log23 and 2.

Keywords
roundness, graph, metric invariant
Mathematical Subject Classification 2000
Primary: 05C99
Secondary: 46B20, 20F65
Milestones
Received: 27 July 2009
Revised: 28 December 2009
Accepted: 29 December 2009
Published: 20 April 2010

Communicated by Scott Chapman
Authors
Matthew Horak
Mathematics, Statistics and Computer Science Department
University of Wisconsin – Stout
Menomonie, WI 54751
United States
http://faculty.uwstout.edu/horakm/
Eric LaRose
Mathematics, Statistics and Computer Science Department
University of Wisconsin – Stout
Menomonie, WI 54751
United States
Jessica Moore
Mathematics, Statistics and Computer Science Department
University of Wisconsin – Stout
Menomonie, WI 54751
United States
Michael Rooney
Mathematics, Statistics and Computer Science Department
University of Wisconsin – Stout
Menomonie, WI 54751
United States
Hannah Rosenthal
Mathematics, Statistics and Computer Science Department
University of Wisconsin – Stout
Menomonie, WI 54751
United States