Results of studies on resolvability and metric dimension in graphs

Abstract

This thesis explores advanced topics in graph theory, focusing on metric dimension, neighbourhood resolving sets, and the @, polynomial across various graph -families and applications. It begins with the analysis of metric dimension and bases in graph composition products, particularly compositions involving star graphs and empty graphs. Graphs such as $¢(0¢), S¢(S;) and other variants are examined for their metric dimension, and the properties of their resolving sets are characterized in depth. Study the detailed analysis of neighbourhood resolving sets in all connected graphs of order at most six, providing insight into the structure and uniqueness of vertex identification based on their neighbourhoods. Further, the neighbourhood resolvability of specific graph classes such as the bipartite graph Ky¢ and graph sums P, + P, , P, + O, are studied extensively. These investigations contribute to understanding the distinct neighborhood structures and resolving capabilities within complex graph constructions. The thesis also delves into the properties of the ®,- polynomial for various graph families, including P, + B, , B+ O, , Kn_e, and the Cartesian product P, X B, .These analyses offer insights into how vertex degrees, distances, and edge distributions shape the structural identity of graphs A novel interdisciplinary approach is presented by modelling the 12 Zodiac signs as graphs and studying their interrelationships using the @,-polynomial, merging mathematical structure with symbolic representations. Another important aspect involves the study of the @,-polynomial and neighborhood resolvability in zero divisor graphs of small finite commutative rings of order at most 10, revealing deep connections between algebraic properties and graph invariants. The practical relevance of neighbourhood resolving sets is highlighted through two real-world applications: (1) selecting a nodal centre for effective drug distribution networks, and (2) optimizing construction labour management by modelling trade-specific labourers as graph vertices. These case studies demonstrate the applicability of graph-theoretic tools to problems in logistics, resource allocation, and operational efficiency. The thesis concludes with a summary of key findings and recommendations for future research in combinatorial optimization and applicd graph theory.