  The smallest set of vertices needed to differentiate or categorize every
other vertex in a graph is referred to as the graph's metric dimension. Finding
the class of graphs for a particular given metric dimension is an NP-hard
problem. This concept has applications in many different domains, including
graph theory, network architecture, and facility location problems. A graph $G$
with order $n$ is known as a Toeplitz graph over the subset $S$ of consecutive
collections of integers from one to $n$, and two vertices will be adjacent to
each 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 a
commutative ring $\mathbb{Z}_{n}$, in which two vertices will be adjacent to
each other if their product will leave the remainder zero under modulo $n$.
Since the local fractional metric dimension problem is NP-hard, it is
computationally difficult to identify an optimal solution or to precisely
determine the minimal size of a local resolving set; in the worst case, the
process takes exponential time. Different upper bound sequences of local
fractional metric dimension are suggested in this article, along with a
comparison analysis for certain families of Toeplitz and zero-divisor graphs.
Furthermore, we note that the analyzed local fractional metric dimension upper
bounds fall into three metric families: constant, limited, and unbounded.
