On randomly -dimensional graphs (original) (raw)
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Characterization of Randomly k-Dimensional Graphs
Arxiv preprint arXiv:1103.3570, 2011
For an ordered set W = {w 1 , w 2 , . . . , w k } of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W ) := (d(v, w 1 ), d(v, w 2 ), . . . , d(v, w k )) is called the (metric) representation of v with respect to W , where d(x, y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W . A resolving set for G with minimum cardinality is called a basis of G and its cardinality is the metric dimension of G. A connected graph G is called randomly k-dimensional graph if each k-set of vertices of G is a basis of G. In this paper, we study randomly k-dimensional graphs and provide some properties of these graphs.
On the Dimension and Euler characteristic of random graphs
The inductive dimension dim(G) of a finite undirected graph G is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the empty graph has dimension −1. We look at the distribution of the random variable dim on the Erdös-Rényi probability space G(n, p), where each of the n(n − 1)/2 edges appears independently with probability 0 ≤ p ≤ 1. We show that the average dimension dn(p) = Ep,n[dim] is a computable polynomial of degree n(n − 1)/2 in p. The explicit formulas allow experimentally to explore limiting laws for the dimension of large graphs. In this context of random graph geometry, we mention explicit formulas for the expectation Ep,n[χ] of the Euler characteristic χ, considered as a random variable on G(n, p). We look experimentally at the statistics of curvature K(v) and local dimension dim(v) = 1 + dim(S(v)) which satisfy χ(G) = v∈V K(v) and dim(G) = 1 |V | v∈V dim(v). We also look at the signature functions f (p) = Ep[dim], g(p) = Ep[χ] and matrix values functions Av,w(p) = Covp[dim(v), dim(w)], Bv,w(p) = Cov[K(v), K(w)] on the probability space G(p) of all subgraphs of a host graph G = (V, E) with the same vertex set V , where each edge is turned on with probability p.
The k-metric dimension of a graph
2015
As a generalization of the concept of a metric basis, this article introduces the notion of kkk-metric basis in graphs. Given a connected graph G=(V,E)G=(V,E)G=(V,E), a set SsubseteqVS\subseteq VSsubseteqV is said to be a kkk-metric generator for GGG if the elements of any pair of different vertices of GGG are distinguished by at least kkk elements of SSS, {\em i.e.}, for any two different vertices u,vinVu,v\in Vu,vinV, there exist at least kkk vertices w1,w2,ldots,wkinSw_1,w_2,\ldots,w_k\in Sw1,w2,ldots,wkinS such that dG(u,wi)nedG(v,wi)d_G(u,w_i)\ne d_G(v,w_i)dG(u,wi)nedG(v,wi) for every iin1,ldots,ki\in \{1,\ldots,k\}iin1,ldots,k. A metric generator of minimum cardinality is called a kkk-metric basis and its cardinality the kkk-metric dimension of GGG. A connected graph GGG is \emph{$k$-metric dimensional} if kkk is the largest integer such that there exists a kkk-metric basis for GGG. We give a necessary and sufficient condition for a graph to be kkk-metric dimensional and we obtain several results on the kkk-metric dimension.
is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we study some properties of uniquely dimensional graphs.
Some properties of the multiset dimension of graphs
Electronic Journal of Graph Theory and Applications
The multiset dimension was introduced by Rinovia Simanjuntak et al. as a variation of metric dimension. In this problem, the representation of a vertex v with respect to a resolving set W is expressed as a multiset of distances between v and all vertices in W , including their multiplicities. The multiset dimension is defined to be the minimum cardinality of the resolving set. Clearly, this is at least the metric dimension of a graph. In this paper, we study the properties of the multiset dimension of graphs.
An introduction to the theory of random graphs
2015
This thesis provides an introduction to the fundamentals of random graph theory. The study starts introduces the two fundamental building blocks of random graph theory, namely discrete probability and graph theory. The study starts by introducing relevant concepts probability commonly used in random graph theory-these include concentration inequalities such as Chebyshev's inequality and Chernoff's inequality. Moreover we proceed by introducing central concepts in graph theory, which will underpin the later discussion. In particular we provide results such as Mycielski's construction of a family of triangle-free graphs with high chromatic number and results in Ramsey theory. Next we introduce the concept of a random graph and present two of the most famous proofs in graph theory using the theory random graphs. These include the proof of the fact that there are graphs with arbitrarily high girth and chromatic number, and a bound on the Ramsey number R(k, k). Finally we conclude by introducing the notion of a threshold function for a monotone graph property and we present proofs for the threhold functions of certain properties.
The dimension and Euler characteristic of random graphs
2011
Abstract. The inductive dimension dim(G) of a finite undirected graph G is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the empty graph has dimension −1. We look at the dis-tribution of the random variable dim on the Erdös-Rényi probability space G(n, p), where each of the n(n − 1)/2 edges appears independently with prob-ability 0 ≤ p ≤ 1. We show that the average dimension dn(p) = Ep,n[dim] is a computable polynomial of degree n(n − 1)/2 in p. The explicit formu-las allow experimentally to explore limiting laws for the dimension of large graphs. In this context of random graph geometry, we mention explicit for-mulas for the expectation Ep,n[χ] of the Euler characteristic χ, considered as a random variable on G(n, p). We look experimentally at the statistics of curvature K(v) and local dimension dim(v) = 1 + dim(S(v)) which satisfy χ(G) = v∈V K(v) and dim(G) =
A note on kkk-metric dimensional graphs
arXiv (Cornell University), 2019
Given a graph G = (V, E), a set S ⊂ V is called a k-metric generator for G if any pair of different vertices of G is distinguished by at least k elements of S. A graph is k-metric dimensional if k is the largest integer such that there exists a k-metric generator for G. This paper studies some bounds on the number k for which a graph is k-metric dimensional.
The k-metric dimension of graphs: a general approach
arXiv (Cornell University), 2016
Let (X, d) be a metric space. A set S ⊆ X is said to be a k-metric generator for X if and only if for any pair of different points u, v ∈ X, there exist at least k points w 1 , w 2 ,. .. w k ∈ S such that d(u, w i) = d(v, w i), for all i ∈ {1,. .. k}. Let R k (X) be the set of metric generators for X. The k-metric dimension dim k (X) of (X, d) is defined as dim k (X) = inf{|S| : S ∈ R k (X)}. Here, we discuss the k-metric dimension of (V, d t), where V is the set of vertices of a simple graph G and the metric d t : V × V → N ∪ {0} is defined by d t (x, y) = min{d(x, y), t} from the geodesic distance d in G and a positive integer t. The case t ≥ D(G), where D(G) denotes the diameter of G, corresponds to the original theory of k-metric dimension and the case t = 2 corresponds to the theory of k-adjacency dimension. Furthermore, this approach allows us to extend the theory of k-metric dimension to the general case of non-necessarily connected graphs.