Combinatorial curvature for planar graphs

Higuchi, Yusuke

doi:10.1002/jgt.10004

Cited by 88 publications

(141 citation statements)

References 9 publications

Supporting

Mentioning

137

Contrasting

Unclassified

Order By: Relevance

“…Condition (3) suggests to work with a purely combinatorial notion. Condition (2) rules out "universal" notions such as Gromov curvature [14,17] (2.5)…”

Section: 4mentioning

confidence: 99%

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Luca

Friesecke

2017

J Nonlinear Sci

View full text Add to dashboard Cite

Abstract. We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem [16], which concerns a system of N identical atoms in two dimensions interacting via the idealized pair potential V (r) = +∞ if r < 1, −1 if r = 1, 0 if r > 1. This is done by endowing the bond graph of a general particle configuration with a suitable notion of discrete curvature, and appealing to a discrete GaussBonnet theorem [18] which, as its continuous cousins, relates the sum/integral of the curvature to topological invariants. This leads to an exact geometric decomposition of the Heitmann-Radin energy into (i) a combinatorial bulk term, (ii) a combinatorial perimeter, (iii) a multiple of the Euler characteristic, and (iv) a natural topological energy contribution due to defects. An analogous exact geometric decomposition is also established for soft potentials such as the Lennard-Jones potential V (r) = r −6 − 2r −12 , where two additional contributions arise, (v) elastic energy and (vi) energy due to non-bonded interactions.

show abstract

“…Condition (3) suggests to work with a purely combinatorial notion. Condition (2) rules out "universal" notions such as Gromov curvature [14,17] (2.5)…”

Section: 4mentioning

confidence: 99%

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Luca

Friesecke

2017

J Nonlinear Sci

View full text Add to dashboard Cite

show abstract

“…The computations are done separately on vertices with valence less than, equal, or greater than six, excluding all boundary vertices. That is, we treat separately vertices that are topologically convex, planar, or concave in the combinatorial Gaussian curvature sense [Higuchi 2001]. The total number of vectors obtained from the mesh vertices is 24.…”

Section: Feature Extractionmentioning

confidence: 99%

Mesh Discriminative Features for 3D Steganalysis

Yang

Ivrissimtzis

2014

ACM Trans. Multimedia Comput. Commun. Appl.

View full text Add to dashboard Cite

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. We propose a steganalytic algorithm for triangle meshes, based on the supervised training of a classifier by discriminative feature vectors. After a normalization step, the triangle mesh is calibrated by one step of Laplacian smoothing and then a feature vector is computed, encoding geometric information corresponding to vertices, edges and faces. For a given steganographic or watermarking algorithm, we create a training set containing unmarked meshes and meshes marked by that algorithm, and train a classifier using Quadratic Discriminant Analysis. The performance of the proposed method was evaluated on six well-known watermarking/steganographic schemes with satisfactory accuracy rates.

show abstract

“…DeVos and Mohar [8] obtained the following Gauss-Bonnet inequality (3.1), which solves a conjecture of Higuchi [11]. Higuchi's conjecture can be thought of as a combinatorial analogue of Myers' theorem [4,14] on Riemannian manifolds.…”

Section: The Characterization Of Embedded Graphs With Nonnegative Commentioning

confidence: 90%

“…For the motivation of Gaussian curvature below, we refer to the work of Allendoerfer and Weil [2], Banchoff [3], Cheeger, Müller, and Schrader [5], McMullen [13], and the present author [6]. For the motivation of combinatorial curvature, we refer to Gromov [10], Higuchi [11], and Ishida [12]. For the application of the Gaussian curvature and the combinatorial curvature on surfaces, we refer to the recent work of DeVos and Mohar [8] and the author's joint work with G. Chen [7].…”

Section: Vertices 1-cells Open Segments and 2-cells Open Convex Polmentioning

confidence: 99%

See 1 more Smart Citation

The Gauss-Bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature

Chen¹

2008

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let M be a connected d-manifold without boundary obtained from a (possibly infinite) collection P of polytopes of R d by identifying them along isometric facets. Let V (M ) be the set of vertices of M . For each v ∈ V (M ), define the discrete Gaussian curvature κ M (v) as the normal angle-sum with sign, extended over all polytopes having v as a vertex. Our main result is as follows: If the absolute total curvature v∈V (M ) |κ M (v)| is finite, then the limiting curvature κ M (p) for every end p ∈ End M can be well-defined and the Gauss-Bonnet formula holds:In particular, if G is a (possibly infinite) graph embedded in a 2-manifold M without boundary such that every face has at least 3 sides, and if the combinatorial curvature Φ G (v) ≥ 0 for all v ∈ V (G), then the number of vertices with nonvanishing curvature is finite. Furthermore, if G is finite, then M has four choices: sphere, torus, projective plane, and Klein bottle. If G is infinite, then M has three choices: cylinder without boundary, plane, and projective plane minus one point. Polytopal manifolds

show abstract

Combinatorial curvature for planar graphs

Cited by 88 publications

References 9 publications

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Mesh Discriminative Features for 3D Steganalysis

The Gauss-Bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature

Contact Info

Product

Resources

About