Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures
We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing graphs with edges take...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2016-02-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/8/3/9 |
| _version_ | 1798041187963109376 |
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| author | Tyler Reese Randy Paffenroth Joseph D. Fehribach |
| author_facet | Tyler Reese Randy Paffenroth Joseph D. Fehribach |
| author_sort | Tyler Reese |
| collection | DOAJ |
| description | We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing graphs with edges taken as vectors. This leads to a discussion of Kirchhoff graphs, as originally presented by Fehribach, which can be defined independent of any matrix relations. In particular, we present simple cases in which vector graphs are guaranteed to be Kirchhoff or non-Kirchhoff. Next, we review Maxwell’s method of drawing reciprocal figures as he presented in 1864, using modern mathematical language. We then demonstrate cases in which R2-vector graphs defined from Maxwell reciprocals are “dual” Kirchhoff graphs. Given an example in which Maxwell’s theories are not sufficient to define vector graphs, we begin to explore other methods of developing dual Kirchhoff graphs. |
| first_indexed | 2024-04-11T22:18:00Z |
| format | Article |
| id | doaj.art-32f30b5d7bd047578d98fcb1311d54e3 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T22:18:00Z |
| publishDate | 2016-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-32f30b5d7bd047578d98fcb1311d54e32022-12-22T04:00:19ZengMDPI AGSymmetry2073-89942016-02-0183910.3390/sym8030009sym8030009Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal FiguresTyler Reese0Randy Paffenroth1Joseph D. Fehribach2Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, USADepartment of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, USADepartment of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, USAWe compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing graphs with edges taken as vectors. This leads to a discussion of Kirchhoff graphs, as originally presented by Fehribach, which can be defined independent of any matrix relations. In particular, we present simple cases in which vector graphs are guaranteed to be Kirchhoff or non-Kirchhoff. Next, we review Maxwell’s method of drawing reciprocal figures as he presented in 1864, using modern mathematical language. We then demonstrate cases in which R2-vector graphs defined from Maxwell reciprocals are “dual” Kirchhoff graphs. Given an example in which Maxwell’s theories are not sufficient to define vector graphs, we begin to explore other methods of developing dual Kirchhoff graphs.http://www.mdpi.com/2073-8994/8/3/9Kirchhoff graphsgeometric graphsdualityMaxwell reciprocals |
| spellingShingle | Tyler Reese Randy Paffenroth Joseph D. Fehribach Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures Symmetry Kirchhoff graphs geometric graphs duality Maxwell reciprocals |
| title | Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures |
| title_full | Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures |
| title_fullStr | Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures |
| title_full_unstemmed | Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures |
| title_short | Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures |
| title_sort | duality in geometric graphs vector graphs kirchhoff graphs and maxwell reciprocal figures |
| topic | Kirchhoff graphs geometric graphs duality Maxwell reciprocals |
| url | http://www.mdpi.com/2073-8994/8/3/9 |
| work_keys_str_mv | AT tylerreese dualityingeometricgraphsvectorgraphskirchhoffgraphsandmaxwellreciprocalfigures AT randypaffenroth dualityingeometricgraphsvectorgraphskirchhoffgraphsandmaxwellreciprocalfigures AT josephdfehribach dualityingeometricgraphsvectorgraphskirchhoffgraphsandmaxwellreciprocalfigures |