301. On the metric dimension and diameter of circulant graphs with three jumps
- Author
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Saima Rashid, Andrea Semaničová-Feňovčíková, Abdul Qudair Baig, and Muhammad Imran
- Subjects
Computer Science::Information Retrieval ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Dimension function ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,0102 computer and information sciences ,01 natural sciences ,Vertex (geometry) ,Metric dimension ,Combinatorics ,Metric space ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Circulant graph ,010201 computation theory & mathematics ,Bounded function ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Computer Science::General Literature ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Circulant matrix ,ComputingMilieux_MISCELLANEOUS ,Connectivity ,Mathematics - Abstract
Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.
- Published
- 2018
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