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The goal of this study is to characterize specific aspects of bipartite and inverse graphs that are associated with specific algebraic structures, such as weak inverse property quasigroups and their isotopes, commutator subloops, associator subloops, and nuclei, with a focus on structural and topological characteristics. This research aims to highlight the link between mathematical algebraic systems and graph\u2010theoretic capabilities, paving the path for theoretical advances and applications in computer science through Simulink. The methodology blends algebraic techniques based on quasigroup structural components with basic ideas of simple graphs via edge labeling. Furthermore, mathematical methods are used for property analysis, graph visualization, and construction. The analysis shows that inverse and bipartite graphs with weak inverse property loops have distinct structural patterns, such as supporting substructures of specific properties, connectedness, and symmetry in the vertex system. 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