Geometric point-circle pentagonal geometries from Moore graphs

On Symmetric Moore Geometries

Moore Geometries ( Part Iv

For the relevant literature and most of the definitions and conventions the reader is referred to [1]. The next section contains an explicit description of the (reduced) characteristic polynomial of L4(s, t), and it is shown that this polynomial admits a decomposition into two closely related factors of degree 2 each. In Section 3 it is first shown that at least one of these factors must be red...

Bounds for point recolouring in geometric graphs

We examine a recolouring scheme ostensibly used to assist in classifying geographic data. Given a drawing of a graph with bi-chromatic vertices, a vertex can be recoloured if it is surrounded by neighbours of the opposite colour. The notion of surrounded is defined as a contiguous subset of neighbours that span an angle greater than 180 degrees. The recolouring of surrounded vertices continues ...

Two-point concentration in random geometric graphs

A random geometric graph Gn is constructed by taking vertices X1, . . . , Xn ∈ R at random (i.i.d. according to some probability distribution ν with a bounded density function) and including an edge between Xi and Xj if ‖Xi−Xj‖ < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r = r(n) is chosen such that nr = o(lnn) then the probability distribution of the cliqu...

Point cloud surfaces using geometric proximity graphs

We present a new definition of an implicit surface over a noisy point cloud, based on the weighted least squares approach. It can be evaluated very fast, but artifacts are significantly reduced. We propose to use a different kernel function that approximates geodesic distances on the surface by utilizing a geometric proximity graph. From a variety of possibilities, we have examined the Delaunay...