We study the asymptotic growth of the diameter of the graph obtained by adding sparse “long” edges to a square box in Z. We focus on the cases when an edge between x and y is added with probability decaying with their Euclidean distance like |x − y| as |x − y| → ∞. For s ∈ (d, 2d) we show that the graph diameter for a box of side L scales like (logL) where ∆ = log2(2d/s). In other words, the di...

Given a planar graph G, the graph theoretic dilation of G is defined as the maximum ratio of the shortest-path distance and the Euclidean distance between any two vertices of G. Given a planar point set S, the graph theoretic dilation of S is the minimum graph theoretic dilation that any triangulation of S can achieve. We study the graph theoretic dilation of the regular n-gon. In particular, w...

Abstract We consider a convolution-type operator on vector bundles over metric-measure spaces. This extends the analogous convolution Laplacian functions in our earlier work to bundles, and is natural extension of graph connection Laplacian. prove that for Euclidean or Hermitian connections closed Riemannian manifolds, spectrum this both approximate

In this paper we begin the classification completed in [12] of all partial linear spaces n , graphs F, and groups G which satisfy one of the following: I. II = (0>, ££) is a connected partial linear space of order 2 in which every pair of intersecting lines lies in a subspace isomorphic to the dual of an affine plane of order 2; II. F is a connected graph such that, for each vertex x of F, the ...

We consider the quantum graph Hamiltonian on square lattice in Euclidean space, and we show that spectrum of converges to corresponding Schr\"odinger operator space continuum limit, eigenfunctions eigenprojections also converge some sense. employ discrete as intermediate operator, use a recent result by second third author limit operator.

PROBLEM DEFINITION Consider a graph G = (V,E). A subset C of V is called a dominating set if every vertex is either in C or adjacent to a vertex in C. If, furthermore, the subgraph induced by C is connected, then C is called a connected dominating set. A connected dominating set with a minimum cardinality is called a minimum connected dominating set (MCDS). Computing a MCDS is an NP-hard proble...

Average-half-face experiments the overall accuracy of the system is better than using the original full face image. Clearly experiment shows that half face data produces higher recognition accuracy [5]. The average-half-face contain the data exactly half of the full face and thus results in storage and computational time saving. The information stored in average-half-face may be more discrimina...

In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show t...

Voronoi diagrams are fundamental data structures in computational geometry with several applications on different fields inside and outside computer science. This paper shows a CUDA algorithm to compute Voronoi diagrams on a 2D image where the distance between points cannot be directly computed in the euclidean plane. The proposed method extends an existing Dijkstra-based GPU algorithm to treat...