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The smallest set of vertices needed to differentiate or categorize everyother vertex in a graph is referred to as the graph's metric dimension. Findingthe class of graphs for a particular given metric dimension is an NP-hardproblem. This concept has applications in many different domains, includinggraph theory, network architecture, and facility location problems. A graph $G$with order $n$ is known as a Toeplitz graph over the subset $S$ of consecutivecollections of integers from one to $n$, and two vertices will be adjacent toeach other if their absolute difference is a member of $S$. A graph$G(\mathbb{Z}_{n})$ is called a zero-divisor graph over the zero divisors of acommutative ring $\mathbb{Z}_{n}$, in which two vertices will be adjacent toeach other if their product will leave the remainder zero under modulo $n$.Since the local fractional metric dimension problem is NP-hard, it iscomputationally difficult to identify an optimal solution or to preciselydetermine the minimal size of a local resolving set; in the worst case, theprocess takes exponential time. Different upper bound sequences of localfractional metric dimension are suggested in this article, along with acomparison analysis for certain families of Toeplitz and zero-divisor graphs.Furthermore, we note that the analyzed local fractional metric dimension upperbounds fall into three metric families: constant, limited, and unbounded.

On Sharp Bounds of Local Fractional Metric Dimension for Certain Symmetrical Algebraic Structure Graphs