On Rotationally Symmetrical Planar Networks and Their Local Fractional Metric Dimension

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 map ϑ:V(G)→[0,1] such that ∑u∈R{v,w}ϑ(u)≥1, for every pair of adjacent distinct vertices v,w∈V(G). The local fractional metric dimension of the graph G is defined as ldimf(G) = min{∑v∈V(G)ϑ(v), where ϑ is a local resolving function of G}. 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.