On rotationally symmetrical planar networks and their local fractional metric dimension

College

College of Science

Department/Unit

Mathematics and Statistics Department

Document Type

Article

Source Title

Symmetry

Volume

15

Issue

2

First Page

530

Publication Date

2023

Abstract

The metric dimension has various applications in several fields, such as computer science, image processing, pattern recognition, integer programming problems, drug discovery, and the production of various chemical compounds. The lowest number of vertices in a set with the condition that any vertex can be uniquely identified by the list of distances from other vertices in the set is the metric dimension of a graph. A resolving function of the graph G is a mapπœ—:𝑉(𝐺)β†’[0,1]such thatβˆ‘π‘’βˆˆβ„›{𝑣,𝑀}πœ—(𝑒)β‰₯1,for every pair of adjacent distinct vertices𝑣,π‘€βˆˆπ‘‰(𝐺). The local fractional metric dimension of the graph G is defined asldimf(𝐺)=min{βˆ‘π‘£βˆˆπ‘‰(𝐺)πœ—(𝑣), whereπœ—is a local resolving function of𝐺}. This paper presents a new family of planar networks namely, rotationally heptagonal symmetrical graphs by means of up to four cords in the heptagonal structure, and then find their upper-bound sequences for the local fractional metric dimension. Moreover, the comparison of the upper-bound sequence for the local fractional metric dimension is elaborated both numerically and graphically. Furthermore, the asymptotic behavior of the investigated sequences for the local fractional metric dimension is addressed.

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Disciplines

Mathematics

Keywords

Graph theory; Symmetry

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