Mersenne banner
no. 1
Curvature on a graph via its geometric spectrum
Paul Baird1
1 Laboratoire de Mathématiques de Bretagne Atlantique Université de Bretagne Occidentale 6 av. Victor Le Gorgeu – CS 93837 29238 BREST CEDEX FRANCE
Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 97-105.

We approach the problem of defining curvature on a graph by attempting to attach a ‘best-fit polytope’ to each vertex, or more precisely what we refer to as a configured star. How this should be done depends upon the global structure of the graph which is reflected in its geometric spectrum. Mean curvature is the most natural curvature that arises in this context and corresponds to local liftings of the graph into a suitable Euclidean space. We discuss some examples.

Published online:
DOI: 10.5802/acirm.59
Classification: 05C10, 52C99, 52B11, 39A14
Keywords: graph theory, curvature, geometric spectrum, shape recognition
Paul Baird 1

1 Laboratoire de Mathématiques de Bretagne Atlantique Université de Bretagne Occidentale 6 av. Victor Le Gorgeu – CS 93837 29238 BREST CEDEX FRANCE 
@article{ACIRM_2013__3_1_97_0,
     author = {Paul Baird},
     title = {Curvature on a graph via its geometric spectrum},
     journal = {Actes des rencontres du CIRM},
     pages = {97--105},
     publisher = {CIRM},
     volume = {3},
     number = {1},
     year = {2013},
     doi = {10.5802/acirm.59},
     zbl = {1462.05107},
     language = {en},
     url = {https://acirm.centre-mersenne.org/articles/10.5802/acirm.59/}
}
TY  - JOUR
AU  - Paul Baird
TI  - Curvature on a graph via its geometric spectrum
JO  - Actes des rencontres du CIRM
PY  - 2013
SP  - 97
EP  - 105
VL  - 3
IS  - 1
PB  - CIRM
UR  - https://acirm.centre-mersenne.org/articles/10.5802/acirm.59/
UR  - https://zbmath.org/?q=an%3A1462.05107
UR  - https://doi.org/10.5802/acirm.59
DO  - 10.5802/acirm.59
LA  - en
ID  - ACIRM_2013__3_1_97_0
ER  - 
%0 Journal Article
%A Paul Baird
%T Curvature on a graph via its geometric spectrum
%J Actes des rencontres du CIRM
%D 2013
%P 97-105
%V 3
%N 1
%I CIRM
%U https://doi.org/10.5802/acirm.59
%R 10.5802/acirm.59
%G en
%F ACIRM_2013__3_1_97_0
Paul Baird. Curvature on a graph via its geometric spectrum. Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 97-105. doi : 10.5802/acirm.59. https://acirm.centre-mersenne.org/articles/10.5802/acirm.59/

[1] P. Baird A class of quadratic difference equations on a finite graph, arXiv:1109.3286 [math-ph]

[2] P. Baird Constant mean curvature polytopes and hypersurfaces via projections, Differential Geom. and its Applications, Volume online version nov. 2013 | MR | Zbl

[3] P. Baird Emergence of geometry in a combinatorial universe, J. Geom. and Phys., Volume 74 ((2013)), pp. 185-195 | DOI | MR | Zbl

[4] P. Baird Information, universality and consciousness: a relational perspective, Mind and Matter, Volume 11(2) ((2013)), pp. 21-43

[5] P. Baird An invariance property for frameworks in Euclidean space, Linear Algebra and its Applications, Volume 440 ((2014)), pp. 243-265 | DOI | MR | Zbl

[6] P. Baird; M. Wehbe Twistor theory on a finite graph, Comm. Math. Phys., Volume 304(2) ((2011)), pp. 499-511 | DOI

[7] S. Barré Real and discrete holomorphy: introduction to an algebraic approach, J. Math. Pures Appl., Volume 87 ((2007)), pp. 495-513 | DOI | MR | Zbl

[8] C. Delaunay Sur la surface de révolution dont la courbure moyenne est constante, Volume 6 ((1841)), pp. 309-320

[9] R. Descartes Progymnasmata de solidorum elementis, Oeuvres de Descartes, Volume X, pp. 265-276 | Zbl

[10] M. G. Eastwood; R. Penrose Drawing with complex numbers, Math. Intelligencer, Volume 22 ((2000)), pp. 8-13 | DOI | MR | Zbl

Cited by Sources: