The Power-O graph of a finite group is a canonical structural representation, that is, a representation of the group whereby each of the group members is represented by a vertex and the relationship between them is defined by the power relationships between the orders of these members. In a finite group G, the graph relates two different elements of different orders in G, i.e. o(x) ≠o(y), when one of these orders is a non-trivial power of the other, i.e. o(x) =o(y)^m. This exploration develops the impact of algebraic framework of G to the seizing graph characteristics. We obtain such basic invariants as connectedness, diameter, multipartiteness and edge bounds of the Power -O graph. In addition, we give algorithmic methods of building such graphs and computational complexity analysis, hence offering a complete toolkit to researchers who want to use such structures in a computational framework. In addition to the theoretical advancement, this paper shows the applicability of Power-O graphs in engineering. These include network topology modelling, communication hierarchy design, distributed systems, and cryptographic constructions that use graph-based primitives, thus demonstrating the applicability of the graph concept in applied areas. Case based studies demonstrates that a specific structure of Power-O graphs is inherently related to hub-based networks, hierarchical multi-layers networks and clustered communication networks, hence highlighting the possibility of such mathematical constructs to reflect rich network behaviours that are seen in practice. Finally, the current research reinforces the conceptual continuum between the group theory and the graph theory to make the Power-O graph a mathematically sound but practical useful model to study, analyze, and optimize complex networked systems.
