Metric Dimension of Nonplanar Networks by Fractional Technique With Application

The fractional versions of graph-theoretic invariants expand the range of applications like connectivity, scheduling, assignment, and operational research. To investigate this fascinating dimension of fractional graph theory, we present the fractional version of the local metric dimension of networks. The local resolving neighborhood $LR(xy)$ of an edge xy in a network $\mathcal {T}$ is the set of vertices in $\mathcal {T}$ that distinguish between the vertices x and y. A function $\rho :V(\mathcal {T})\rightarrow [{0,1}]$ is considered a local resolving function of $\mathcal {T}$ if $\rho (L(xy))\geq 1$ for all edges xy in $\mathcal {T}$ . The fractional local metric dimension of $\mathcal {T}$ is the minimum value of $\rho (V(\mathcal {T}))$ across all local resolving functions $\rho $ of $\mathcal {T}$ . In this article, we primarily calculate the local-based fractional metric dimension by finding precise values for the non-planar connected network with the name of subdivision of wheel network $AWW(n, k)$ . Additionally, we investigate an application involving the automation of washing machine functions to demonstrate the practical applicability of our findings.