Fractional Metric Dimension of Generalized Sunlet Networks

Abstract

Let N = V N , E N be a connected network with vertex V N and edge set E N ⊆ V N , E N . For any two vertices a and b , the distance d a , b is the length of the shortest path between them. The local resolving neighbourhood (LRN) set for any edge e = ab of N is a set of all those vertices whose distance varies from the end vertices a and b of the edge e . A real-valued function Φ from V N to 0,1 is called a local resolving function (LRF) if the sum of all the labels of the elements of each LRN set remains greater or equal to 1. Thus, the local fractional metric dimension (LFMD) of a connected network N is dim lf N = min Φ : Φ is minimal LRF of N . In this study, LFMD of various types of sunlet-related networks such as sunlet network ( S m ), middle sunlet network ( MS m ), and total sunlet network ( TS m ) are studied in the form of exact values and sharp bounds under certain conditions. Furthermore, the unboundedness and boundedness of all the obtained results of LFMD of the sunlet networks are also checked.

Metric-Based Fractional Dimension of Rotationally-Symmetric Line Networks

“…The lower bound of the LFMD is improved by Javaid et al and they also established the bounds of the LFMD of antiprism and sun flower networks [26]. Since discovering the bounds of the LFMD of generalized sunlet [27] and convex polytopes, [28,29], Sierpinski networks have been established [30]. Now, we are presenting some applications of MD in the field of chemical graph theory; the chemical graph theory applies in chemistry and focuses on the molecular topology.…”

Metric Based Fractional Dimension of Toeplitz Networks

“…Recently, Javaid et al [13] improved the lower bound of LFMD from unity and they also established a computational cetraria to compute exact values of LFMD of the connected networks under certain conditions . Upper and lower bounds of LFMD of generalized gear, convex polytopes and sunlet networks, we refer ' [29,17], [15]'. In this paper, we have computed LFMD of Toeplitz networks T m 1, 2 and T m 1, 4 in the form sharp bounds.…”

Studies of Connected Networks via Fractional Metric Dimension

“…FMD of vertex transitive and hierarchical product of networks was computed by Feng et al Moreover, they also established bounds of FMD of Cartesian product of networks [18,19]. For the study of FMD of corona product, lexicographic product, and unicyclic and generalized sunlet networks, see [20][21][22]. Later on Al Khalidi et al designed computational criteria to compute bounds of FMD of connected networks [23].…”