Gauss-Bonnet Formula, Finiteness Condition, and Characterizations of Graphs Embedded in Surfaces

Chen, Beifang; Chen, Guantao

doi:10.1007/s00373-008-0782-z

Cited by 31 publications

(99 citation statements)

References 6 publications

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“…We refer to [2] for background and proof in the case of tessellations, see also [4,9] for further reference.…”

Section: Definitions and A Combinatorial Gauss-bonnet Formulamentioning

confidence: 99%

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Curvature, Geometry and Spectral Properties of Planar Graphs

Keller

2011

Discrete Comput Geom

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We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally similar to a tessellation. We use this to extend several results known for tessellations to general planar graphs. For non-positive curvature, we show that the graph admits no cut locus and we give a description of the boundary structure of distance balls. For negative curvature, we prove that the interiors of minimal bigons are empty and derive explicit bounds for the growth of distance balls and Cheeger's constant. The latter are used to obtain lower bounds for the bottom of the spectrum of the discrete Laplace operator. Moreover, we give a characterization for triviality of the essential spectrum by uniform decrease of the curvature. Finally, we show that non-positive curvature implies the absence of finitely supported eigenfunctions for nearest neighbor operators.

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“…We refer to [2] for background and proof in the case of tessellations, see also [4,9] for further reference.…”

Section: Definitions and A Combinatorial Gauss-bonnet Formulamentioning

confidence: 99%

“…In contrast to the statement about the infinity, positive curvature implies finiteness of the graph. This was studied in [4,9,15,25,26]. Note also that locally tessellating graphs, recently introduced in [20] (see also [30]), allow for a unified treatment of planar tessellations and trees.…”

Section: Introductionmentioning

confidence: 99%

Curvature, Geometry and Spectral Properties of Planar Graphs

Keller

2011

Discrete Comput Geom

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“…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%

“…To obtain some finiteness results on discrete curvature, Chen and Chen [7] introduced absolute total curvature, and obtained the following relation between the finiteness of total curvature and the finiteness of the number of vertices of nonvanishing curvature. Higuchi's conjecture is modified by Chen and Chen [7] as follows. In what follows we shall see that Theorems 3.1 and 3.3 imply the following characterization of regular polygonal surfaces with nonnegative curvature.…”

Section: Conjecture 32 (Higuchi [11]) Let G Be a (Possibly Infinitementioning

confidence: 99%

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

Chen¹

2008

Proc. Amer. Math. Soc.

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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

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“…In case of finite graphs, the Gauss‐Bonnet theorem reads as (see, eg, )