Fractional Metric Dimension of Generalized Sunlet Networks

Abstract

Let $N=\left(V\left(N\right),E\left(N\right)\right)$ be a connected network with vertex $V\left(N\right)$ and edge set $E\left(N\right)\subseteq \left(V\left(N\right),E\left(N\right)\right)$ . For any two vertices $a$ and $b$ , the distance $d\left(a,b\right)$ is the length of the shortest path between them. The local resolving neighbourhood (LRN) set for any edge $e=\text{ab}$ 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 $\mathrm{\Phi }$ from $V\left(N\right)$ to $\left[\mathrm{0,1}\right]$ 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 ${\dim }_{\text{lf}}\left(N\right)=\min \left\{\left|\mathrm{\Phi }\right|:\mathrm{\Phi }\text{is\hspace{0.17em}minimal\hspace{0.17em}LRF\hspace{0.17em}of}N\right\}$ . In this study, LFMD of various types of sunlet-related networks such as sunlet network ( $S_m$ ), middle sunlet network ( ${\text{MS}}_m$ ), and total sunlet network ( ${\text{TS}}_m$ ) 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.