On the fractional metric dimension of corona product graphs and lexicographic product graphs

TL;DR

This paper investigates the fractional metric dimension of corona and lexicographic product graphs, reducing the problem to computing parameters of the component graphs, which aids in understanding their resolving functions.

Contribution

It provides a method to compute the fractional metric dimension of complex product graphs based on simpler parameters of their factors.

Findings

Reduced the computation of fractional metric dimension to factor graph parameters

Established formulas for corona product graphs

Established formulas for lexicographic product graphs

Abstract

A vertex $x$ in a graph $G$ resolves two vertices $u$, $v$ of $G$ if the distance between $u$ and $x$ is not equal to the distance between $v$ and $x$. A function $g$ from the vertex set of $G$ to $[0,1]$ is a resolving function of $G$ if $g(R_G\{u,v\})\geq 1$ for any two distinct vertices $u$ and $v$, where $R_G\{u,v\}$ is the set of vertices resolving $u$ and $v$. The real number $\sum_{v\in V(G)}g(v)$ is the weight of $g$. The minimum weight of all resolving functions for $G$ is called the fractional metric dimension of $G$, denoted by $\dim_f(G)$. In this paper we reduce the problem of computing the fractional metric dimension of corona product graphs and lexicographic product graphs, to the problem of computing some parameters of the factor graphs.