Asim Nadeem2025-11-282025-11-282023-03-07https://escholar.umt.edu.pk/handle/123456789/13518Let Λ = {A1, A2, . . . , Am} be an ordered m−partition of a connected graph G(V (G), E(G)). The partition representation of a node v with respect to Λ is the m-vector r(v|Λ) = (d(v, A1), d(v, A2), . . . , d(v, Am)), where d(v, Ai) = min{d(v, x)|x ∈ Ai, 1 ≤ i ≤ m} is the distance between v and Ai. If the representation vectors r(v|Λ) are distinct for each node of the graph, then partition is called a resolving partition (RP) of G. A RP of least size is known as partition basis (PB) and its size is known as the partition dimension (PD) of G. If the representation vectors r(v|Λ) are distinct for each node of the graph, in at least k places, then the partition is known as the k−partition generator (k−PG) of G. A k−partition generator of least size is known as k−partition basis (k−PB) and size of Λ is known as the k−partition dimension (k−PD) of G. In essence k−PD can be regarded as the generalization of the concept of PD as every k−PB is also a PB. The localized version of RP, known as local partition generator (LPG), distinguishes the representation vectors r(v|Λ) for only adjacent nodes. The LPG with least size is called local partition basis (LPB) and its size is known as the local partition dimension (LPD) of G. It can also be inferred from the nature of PD that every PB is also an LPB. In this thesis, it is shown that the 2−PD of r−regular graphs is bounded below by 4 for r ≥ 3. It is classified that the 2−PD of graphs of order, n ≥ 4, containing a node of degree at least 4 is bounded below by 4. Further, two algorithms have been designed which can be used in aid of mathematical simulation tools like MATLAB to compute LPB and LPG of finite graphs. We have investigated several important classes of graphs for k−PD when k = 1, 2. Additionally, the LPD is also discussed for some important classes of graphs. In this regard, we have computed PD and 2−PD of prism graphs, n−sunlet graphs, circulant graphs Cn(1, 2), oxide interconnection networks and induced subgraphs of certain hydrocarbon nanotubes. We have also explored the 2−PD of certain families of Toeplitz networks and well known convex polytopes Rn, Qn, Sn, Tn and Dn. The LPD of circulant graphs Cn(1, 2) and convex polytopes Rn, Sn and Tn have also been computed. The applications of the PD related notions in several network optimization problems have also been furnished in the thesis which include delivery services, locating critical patients, unique addressing scheme for oxide interconnection networks to aid routing algorithms, location of warehouses and optimizing the existing setup of facility locations in a certain locality to expedite its services.enThesis