An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture

Abstract

Abstract. Let R be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let V be the set of vertices, and for every v ∈ V , let κ(v) denote the (Gaussian) curvature of v: 2π minus the sum of incident polygon angles. Descartes showed that v∈V κ(v) = 4π whenever R may be realized as the surface of a convex polytope in R 3 . More generally, if R is made of finitely many polygons, Euler's formula is equivalent to the equation v∈V κ(v) = 2πχ(R) where χ(R) is the Euler characteristic of R. Our main theorem shows that whenever v∈V :κ(v)<0 κ(v) converges and there is a positive lower bound on the distance between any pair of vertices in R, there exists a compact closed 2-manifold S and an integer t so that R is homeomorphic to S minus t points, and further v∈V κ(v) ≤ 2πχ(S) − 2πt.In the special case when every polygon is regular of side length one and κ(v) > 0 for every vertex v, we apply our main theorem to deduce that R is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless R is a prism, antiprism, or the projective planar analogue of one of these that |V | ≤ 3444. This resolves a recent conjecture of Higuchi.

Scaled Gromov hyperbolic graphs

“…Because Riemannian manifolds of positive sectional curvature κ uniformly bounded from below as κ(m) ≥ κ min > 0, ∀m ∈ M, have bounded diameter, and because graphs with positive local combinatorial curvature are finite [9,24,7], it follows that, at large scale, only nonpositive curvature is relevant. To define such a concept, let (X, d) be a geodesic metric space.…”

“…First, it is easily observed that the complete graph is positively curved by the intuitive clustering coefficient definition of [8]. From a more precise standpoint, as a consequence of [7,Theorem 1.7], K n has a 2-cell embedding in either S 2 of PR 2 . As far as isometric embedding is concerned, the following can be said (see [15]): It turns out that K n is also positively curved by our definition: indeed, it is readily verified that ∀ K n ,…”

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

“…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.…”

“…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].…”

Generalized Ricci curvature and the geometry of graphs

“…We then have the Gauss-Bonnet formula of G of [9], We can also compare the combinatorial curvature with another version of curvature naturally obtained from the surface S(G), its generalized sectional (Gaussian) curvature. It turns out that the semiplanar graph G has nonnegative combinatorial curvature precisely if the polygonal surface S(G) is an Alexandrov space with nonnegative sectional curvature, i.e.…”