Metric dimension analysis of single-walled titania nanotubes: A graph-theoretic approach

Abstract

Titania nanotubes (TNTs) have attracted growing interest in materials science due to their exceptional properties and broad applications in nanotechnology, catalysis, electronics, and biomedical systems. In this work, we study TNTs through the lens of graph theory by modeling their molecular structures as mathematical graphs. Specifically, we investigate the metric dimension, a key graph invariant that captures how uniquely the positions of all vertices in a graph can be determined using the shortest distances to a selected set of reference vertices, called a resolving set. Two structural models of single-walled TNTs are considered: a three-layered model, denoted by TNT3[m,n], and a six-layered model, denoted by TNT6[m,n], where m and n represent the numbers of titanium atoms along defined lattice directions. We analyze the connectivity and symmetry of these graph representations and derive results for their metric dimension by identifying minimal resolving sets. Our findings offer insight into how structural complexity and layering impact the resolving power of TNT graphs. By connecting mathematical descriptors with nanostructural models, this study contributes to the theoretical understanding of TNTs and provides a potential tool for molecular identification, structure-property prediction, and nanoscale design in chemistry and materials science.