Let M = {v1, v2 ... vn} be an ordered set of minimum number of vertices in a graph G. Then the vector of distances (d(u, v1), d(u, v2) ... d(u, vn)) is called the M -coordinates of a vertex u of G. The set M is called a metric basis if the vertices of G have distinct M -coordinates. The cardinality of M is called Metric dimension of G.
A graph G is said to be (uniquely/ doubly/ triply ) k-dimensional if G has exactly only (one /two/ three) metric bas(i /e)s. In our research journey, we found some family of graphs have exactly 3 metric bases of cardinality 4(Triply 4-dimensional). We are interested to know about some known properties in the literature.
Here are some instances of graphs that have unique dimensional properties: (i) A line graph is an example of a graph with only one dimension. It consists of vertices (points) connected by edges (lines) in a linear pattern, creating a single dimension. (ii) Doubly-dimensional graph: A rectangular arrangement graph is a type of doubly-dimensional graph. It may be represented as vertices placed in a grid-like arrangement, with each vertex linked to its adjacent vertices to form a two-dimensional plane.
(iii) Triply-dimensional graph: A cubic graph is an example of a three-dimensional graph. It may be created as a network of vertices and edges arranged in three dimensions, with each vertex connected to its neighbouring vertices, resulting in a three-dimensional structure.
Mr.Mehboob , This question is not about physical dimension. Kindly read the supporting definitions given.
- Badges
- Science topic
- Similar topics
- Books