Small components in k-nearest neighbour graphs

Explicit laws of large numbers for random nearest-neighbour type graphs

We give laws of large numbers (in the Lp sense) for the total length of the k-nearest neighbours (directed) graph and the j-th nearest neighbour (directed) graph in Rd, d ∈ N, with power-weighted edges. We deduce a law of large numbers for the standard nearest neighbour (undirected) graph. We give the limiting constants, in the case of uniform random points in (0, 1)d, explicitly. Also, we give...

Equitable neighbour-sum-distinguishing edge and total colourings

With any (not necessarily proper) edge k-colouring γ : E(G) −→ {1, . . . , k} of a graph G, one can associate a vertex colouring σγ given by σγ(v) = ∑ e∋v γ(e). A neighbour-sumdistinguishing edge k-colouring is an edge colouring whose associated vertex colouring is proper. The neighbour-sum-distinguishing index of a graph G is then the smallest k for which G admits a neighbour-sum-distinguishin...

Percolation, connectivity, coverage and colouring of random geometric graphs

In this review paper, we shall discuss some recent results concerning several models of random geometric graphs, including the Gilbert disc model Gr , the k-nearest neighbour model G nn k and the Voronoi model GP . Many of the results concern finite versions of these models. In passing, we shall mention some of the applications to engineering and biology.

Optimal weighted nearest neighbour classifiers

We derive an asymptotic expansion for the excess risk (regret) of a weighted nearest-neighbour classifier. This allows us to find the asymptotically optimal vector of nonnegative weights, which has a rather simple form. We show that the ratio of the regret of this classifier to that of an unweighted k-nearest neighbour classifier depends asymptotically only on the dimension d of the feature vec...

On the Connectivity of Random k-th Nearest Neighbour Graphs