A set S of vertices in a graph G is a resolving set if for every pair of vertices u, v ∈ V (G) there is a vertex x ∈ S such that the distances d(x, v) 6= d(x, u). We define a new parameter res(G), the size of the smallest subset S of V (G) that is not a resolving set but every superset of S resolves G. We also demonstrate that for every triple (a, b, c), a ≤ (b + 1) ≤ c, there is a graph G in w...

In this paper we introduce a new notion of generalized metric, called i-metric. This generalization is made by changing the valuation space of the distance function. The result is an interesting distance function for the set of fuzzy numbers of Interval Type with non negative fuzzy numbers as values. This example of i-metric generates a topology in a very natural way, based on open balls. We pr...

Given a graph G, we say S ⊆ V (G) is resolving if for each pair of distinct u, v ∈ V (G) there is a vertex x in S where d(u, x) 6= d(v, x). The metric dimension of G is the minimum cardinality of all resolving sets. For w ∈ V (G), the distance from w to S, denoted d(w, S), is the minimum distance between w and the vertices of S. Given P = {P1, P2, . . . , Pk} an ordered partition of V (G) we sa...

Making an extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorph...

The power graphPG of a finite group G is the graph with the vertex set G, where two distinct vertices are adjacent if one is a power of the other. We first show that PG has a transitive orientation, so it is a perfect graph and its core is a complete graph. Then we use the poset on all cyclic subgroups of G (under usual inclusion) to characterize the structure ofPG. Finally, a closed formula fo...

Minkowski dimension is one of the most widely used dimensions. Its popularity is largely due to its relative ease of mathematical calculation. The definition goes back at least to the 1930’s and it has been variously termed Kolmogorov entropy, metric dimension, information dimension, ... etc. Let F be any non-empty bounded subset of R and let Nγ(F ) be the smallest number of sets of diameter at...

An embedding of one metric space (X, d) into another (Y, ρ) is an injective map f : X → Y . The central genre of problems in the area of metric embedding is finding such maps in which the distances between points do not change “too much”. Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, st...

Let , be a graph with vertex set and edge set . Let then is said to be a local metric basis of , if for any two adjacent vertices , ⁄ , there exists a such that , , . The minimum cardinality of local metric basis is called the local metric dimension (lmd) of graph G. In this paper we investigate the local metric basis and local metric dimension of Cyclic Split Graph .

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...