Any symmetric affinity function w : V × V → R+ defined on a discrete set V induces Euclidean space structure on V . In particular, an undirected graph specified by an affinity (or adjacency ) matrix can be considered as a metric topological space. We have calculated the visual representations of the probabilistic locus for a chain, a polyhedron, and a finite 2-dimensional lattice.

The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found applications in optimization, navigation, network theory, image processing, pattern recognition etc. Several other authors have studied metric dimension of various standard graphs. In this paper we introduce a real valued function called generalized metric + → × × R X X X Gd : where = = ) / ( W v...

For a large class of metric spaces X including discrete groups we prove that the asymptotic Assouad-Nagata dimension AN-asdim X of X coincides with the covering dimension dim(ν L X) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim(X × R) = AN-asdim X + 1. We note that the similar equality for Gromov's asymptoti...

A resolving set of a graph G is a set S ⊆ V (G), such that, every pair of distinct vertices of G is resolved by some vertex in S. The metric dimension of G, denoted by β(G), is the minimum cardinality of all the resolving sets of G. Shamir Khuller et al. [10], in 1996, proved that a graph G with β(G) = 2 can have neither K5 nor K3,3 as its subgraph. In this paper, we obtain a forbidden subgraph...