Abstract
Objective: To explore the algebraic structures inherent in groups and their relationship to graph theory, particularly focusing on the process of converting a group into a graph. Method: Let be a group. The Order Prime Graph of is a graph with and two distinct vertices and are adjacent in if and only if In this paper, we characterize the structure of order prime graphs of some groups and also we find the independence number and clique number of the order prime graph. Finding: In the present article, we characterized the structure of order prime graphs of some groups and also we found the independence number and clique number of the order prime graph. Novelty: Introducing innovative methods for converting groups into graphs, departing from existing approaches, could be a key novelty. This might involve novel ways of encoding group elements as vertices and group operations as edges, potentially leveraging insights from other mathematical fields or computational techniques. Keywords: Order Prime graph, Finite group, Clique number, Independence number, Prime number
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