For an ordered subset W={w1,w2,…,wk} of vertices and a vertex v in connected graph G, the k-vector r(v|W)=(d(v,w1),d(v,w2),…,d(v,wk)) is called representation with respect to W, where d(v,wi) distance between wi, for 1≤i≤k. The set W resolving G if r(u|W)≠r(v|W), every pair u,v∈V(G). minimum positive integer k which has cardinality metric dimension denoted as dim(G). A dim(G) basis G. bipartite...

The spectrum of a matrix M is the multiset that contains all the eigenvalues of M. If M is a matrix obtained from a graph G, then the spectrum of M is also called the graph spectrum of G. If two graphs has the same spectrum, then they are cospectral (or isospectral) graphs. In this paper, we compare four spectra of matrices to examine their accuracy in protein structural comparison. These four ...

An `2 metric is a metric ρ such that √ ρ can be embedded isometrically into R endowed with Euclidean norm, and the minimal possible d is the dimension associated with ρ. A dimension reduction of an `2 metric ρ is an embedding of ρ into another ` 2 2 metric μ so that distances in μ are similar to those in ρ and moreover, the dimension associated with μ is small. Much of the motivation in investi...

Abstract Inspired by the notion of action convergence in graph limit theory, we introduce a measure-theoretic representation matrices, and use it to define new pseudo-metric on space matrices. Moreover, show that such is metric subspace adjacency or Laplacian matrices for graphs. Hence, particular, obtain isomorphism classes Additionally, study how some properties graphs translate this measure ...

A set of vertices W resolves a graph G if every vertex of G is uniquely determined by its vector of distances to the vertices in W . The metric dimension for G, denoted by dim(G), is the minimum cardinality of a resolving set of G. In order to study the metric dimension for the hierarchical product G2 2 uG1 1 of two rooted graphs G2 2 and G u1 1 , we first introduce a new parameter, the rooted ...

It is known that dimension of a set in a metric space can be characterized in information-related terms – in particular, in terms of Kolmogorov complexity of different points from this set. The notion of Kolmogorov complexity K(x) – the shortest length of a program that generates a sequence x – can be naturally generalized to conditional Kolmogorov complexity K(x : y) – the shortest length of a...

An `2 metric is a metric ρ such that √ ρ can be embedded isometrically into R endowed with Euclidean norm, and the minimal possible d is the dimension associated with ρ. A dimension reduction of an `2 metric ρ is an embedding of ρ into another `2 metric μ so that distances in μ are similar to those in ρ and moreover, the dimension associated with μ is small. Much of the motivation in investigat...

Abstract. In this article, we define the transport dimension of probability measures on Rm using ramified optimal transportation theory. We show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called “the dimensional distance,” on the space of probability measu...

In [R.F. Bailey, K. Meagher, On the metric dimension of Grassmann graphs, arXiv:1010.4495 ], Bailey and Meagher obtained an upper bound on the metric dimension of Grassmann graphs. In this note we show that qn+d−1+⌊ d+1 n ⌋ is an upper bound on the metric dimension of bilinear forms graphs Hq(n, d)when n ≥ d ≥ 2. As a result, we obtain an improvement on Babai’s most general bound for the metric...