A study on local fractional metric dimension upper limits for special classes of ϖ-exponent graphs

The concept of metric dimension plays a vital role in numerous fields such as computer science, image processing, pattern recognition, integer programming, pharmaceutical research, and chemical production. The metric dimension of a graph represents the smallest set of vertices capable of uniquely identifying all other vertices based on their distances. However, determining the local fractional metric dimension (LFMD) remains a challenging task due to its computational complexity. In this study, we explore the LFMD of various subclasses of [Formula: see text] graphs, including fan, path, Toeplitz, cycle, and zero-divisor graphs. We derive and analyze the upper bounds of these structures, establishing general patterns for their LFMDs. Our results reveal that the upper bound for [Formula: see text] graphs is [Formula: see text]; for [Formula: see text] it is 2; for [Formula: see text] it is [Formula: see text]; for [Formula: see text] it is 1 when [Formula: see text] and [Formula: see text] is even, and [Formula: see text] is [Formula: see text] when [Formula: see text] and [Formula: see text] is odd; otherwise, it is always [Formula: see text]; and for [Formula: see text] it is [Formula: see text]. These findings contribute to a deeper understanding of the structural properties of [Formula: see text] graphs and open new directions for their applications in complex network modeling and computational analysis.