Linking Bipartiteness and Inversion in Algebra via Graph-Theoretic Methods and Simulink

Research for decades has concentrated on graphs of algebraic structures, which integrate algebra and combinatorics in an innovative way. 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-theoretic 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. Finally, our findings lay the groundwork for future detection of more complex algebraic structures and dynamic graph models, as well as various opportunities for both theoretical research and practical application.