2,634 research outputs found
Metric Dimension of Amalgamation of Graphs
A set of vertices resolves a graph if every vertex is uniquely
determined by its vector of distances to the vertices in . The metric
dimension of is the minimum cardinality of a resolving set of .
Let be a finite collection of graphs and each
has a fixed vertex or a fixed edge called a terminal
vertex or edge, respectively. The \emph{vertex-amalgamation} of , denoted by , is formed by taking all
the 's and identifying their terminal vertices. Similarly, the
\emph{edge-amalgamation} of , denoted by
, is formed by taking all the 's and identifying
their terminal edges.
Here we study the metric dimensions of vertex-amalgamation and
edge-amalgamation for finite collection of arbitrary graphs. We give lower and
upper bounds for the dimensions, show that the bounds are tight, and construct
infinitely many graphs for each possible value between the bounds.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on
Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised
version 21 December 201
The Metric Dimension of Amalgamation of Cycles
For an ordered set W = {w_1, w_2 , ..., w_k } of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1), d(v,w_2 ), ..., d (v,w_k )), where d(x,y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by dim(G), is the number of vertices in a basis of G. Let {G_i} be a finite collection of graphs and each G_i has a fixed vertex voi called a terminal. The amalgamation Amal {Gi , v_{oi}} is formed by taking all of the G_i’s and identifying their terminals. In this paper, we determine the metric dimension of amalgamation of cycles
On the Metric Dimension of Corona Product of Graphs
For an ordered set W = {w_1,w_2,...,w_k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1),d(v,w_2),...,d(v,w_k)) where d(x,y) represents the distance between the vertices x and y. The set W is
called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The metric dimension of G, denoted by dim(G), is the number of vertices in a basis of G. A graph G corona H, G ⊙ H,
is de�fined as a graph which formed by taking n copies of graphs H_1,H_2,...,H_n of H and connecting i-th vertex of G to the vertices of H_i. In this paper, we determine the metric dimension of corona product graphs G⊙H, the lower bound of the metric dimension of K_1 +H and determine some exact values of the metric dimension of G⊙H for some
particular graphs H
Configurational statistics of densely and fully packed loops in the negative-weight percolation model
By means of numerical simulations we investigate the configurational
properties of densely and fully packed configurations of loops in the
negative-weight percolation (NWP) model. In the presented study we consider 2d
square, 2d honeycomb, 3d simple cubic and 4d hypercubic lattice graphs, where
edge weights are drawn from a Gaussian distribution. For a given realization of
the disorder we then compute a configuration of loops, such that the
configurational energy, given by the sum of all individual loop weights, is
minimized. For this purpose, we employ a mapping of the NWP model to the
"minimum-weight perfect matching problem" that can be solved exactly by using
sophisticated polynomial-time matching algorithms. We characterize the loops
via observables similar to those used in percolation studies and perform
finite-size scaling analyses, up to side length L=256 in 2d, L=48 in 3d and
L=20 in 4d (for which we study only some observables), in order to estimate
geometric exponents that characterize the configurations of densely and fully
packed loops. One major result is that the loops behave like uncorrelated
random walks from dimension d=3 on, in contrast to the previously studied
behavior at the percolation threshold, where random-walk behavior is obtained
for d>=6.Comment: 11 pages, 7 figure
Data-Collection for the Sloan Digital Sky Survey: a Network-Flow Heuristic
The goal of the Sloan Digital Sky Survey is ``to map in detail one-quarter of
the entire sky, determining the positions and absolute brightnesses of more
than 100 million celestial objects''. The survey will be performed by taking
``snapshots'' through a large telescope. Each snapshot can capture up to 600
objects from a small circle of the sky. This paper describes the design and
implementation of the algorithm that is being used to determine the snapshots
so as to minimize their number. The problem is NP-hard in general; the
algorithm described is a heuristic, based on Lagriangian-relaxation and
min-cost network flow. It gets within 5-15% of a naive lower bound, whereas
using a ``uniform'' cover only gets within 25-35%.Comment: proceedings version appeared in ACM-SIAM Symposium on Discrete
Algorithms (1998
Introducing Quantum Ricci Curvature
Motivated by the search for geometric observables in nonperturbative quantum
gravity, we define a notion of coarse-grained Ricci curvature. It is based on a
particular way of extracting the local Ricci curvature of a smooth Riemannian
manifold by comparing the distance between pairs of spheres with that of their
centres. The quantum Ricci curvature is designed for use on non-smooth and
discrete metric spaces, and to satisfy the key criteria of scalability and
computability. We test the prescription on a variety of regular and random
piecewise flat spaces, mostly in two dimensions. This enables us to quantify
its behaviour for short lattices distances and compare its large-scale
behaviour with that of constantly curved model spaces. On the triangulated
spaces considered, the quantum Ricci curvature has good averaging properties
and reproduces classical characteristics on scales large compared to the
discretization scale.Comment: 43 pages, 27 figure
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