On Metric Dimension - Edge Metric Dimension and Topological Properties of Connected Graphs

Abstract

The distance between any two vertices of a connected graph is the length of a shortest path between them. Any two vertices/edges are said to be resolved by a vertex if they have different distances w.r.t. that vertex. A set is a resolving set/edge resolving set if every vertex/edge of is resolved by some vertices of , respectively. A topological index is a real number associated to a graph that describes its topological properties. The main objective of this thesis is to study edge metric dimension of certain families of wheel related graphs and generalized Petersen graphs . It has been proved that the gear graph, the fan graph, the friendship graph, the sunflower graph and certain convex polytopes have unbounded edge metric dimension whereas the generalized Petersen graph has a constant edge metric dimension. An explicit Matlab program has also been given to generate the edge resolving set and edge coding for generalized Petersen graphs A comparative study between the metric dimension and the edge metric dimension of the aforementioned families of graphs has also been conducted. The second part of the thesis studies the Zagreb and co-Zagreb indices of - iterated strong double graphs and some eccentricity-based indices of Sierpinski graphs and their regularizations. Moreover, some explicit algorithms have also been given to verify the validity of the results.