منابع مشابه
Note on Metric Dimension
The metric dimension of a compact metric space is defined here as the order of growth of the exponential metric entropy of the space. The metric dimension depends on the metric, but is always bounded below by the topological dimension. Moreover, there is always an equivalent metric in which the metric and topological dimensions agree. This result may be used to define the topological dimension ...
متن کاملThe metric dimension and girth of graphs
A set $Wsubseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,vin V(G)$ there exists $win W$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $dim(G)$. In this paper, it is proved that in a connected graph $...
متن کاملOn the metric dimension and fractional metric dimension for hierarchical product of graphs
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 ...
متن کاملOn the metric dimension of convex polytopes ∗
Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs Gn : F = (Gn)n≥1 depending on n as follows: the order |V (G)| = φ(n) and lim n→∞ φ(n) = ∞ . If there exists a constant C > 0 such that dim(Gn) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension. If all graphs in F hav...
متن کاملOn the metric dimension of Grassmann graphs
The metric dimension of a graph Γ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph Gq(n,k) (whose vertices are the k-subspaces of Fq, and are adjacent if they intersect in a (k− 1)-subspace) for k ≥ 2. We find an upper bound on its metric dimension, which is ...
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ژورنال
عنوان ژورنال: Proceedings of the Japan Academy, Series A, Mathematical Sciences
سال: 1960
ISSN: 0386-2194
DOI: 10.3792/pja/1195524002