Distance-Based Fractional Dimension of Certain Wheel Networks

Abstract

Metric dimension is one of the distance-based parameters which are used to find the position of the robot in a network space by utilizing lesser number of notes and minimum consumption of time. It is also used to characterize the chemical compounds. The metric dimension has a wide range of applications in the field of computer science such as integer programming, radar tracking, pattern recognition, robot navigation, and image processing. A vertex $x$ in a network $\mathbb{W}$ resolves the adjacent pair of vertices $uv$ if $x$ attains an unequal distance from end points of $uv$. A local resolving neighbourhood set $R_L\left(uv\right)$ is a set of vertices of $\mathbb{W}$ which resolve $uv$. A mapping $\alpha :V\left(\mathbb{W}\right)⟶\left[0,1\right]$ is called local resolving function of $\mathbb{W}$ if $\alpha \left(R_L\left(uv\right)\right)\ge 1$ for any adjacent pair of vertices of $uv$ of $\mathbb{W}$ and the minimal value of $\alpha \left(R_L\left(uv\right)\right)$ for all local resolving functions $\alpha$ of $\mathbb{W}$ is called local fractional metric dimension of $\mathbb{W}$. In this paper, we have studied the local fractional metric dimension of wheel-related networks such as web-wheel network, subdivision of wheel network, line network of subdivision of wheel network, and double-wheel network and also examined their boundedness.