Graph-theoretic approach to analyze simple electrical network topology

The mathematical analysis of electrical networks has evolved substantially since Gustav Kirchhoff first established the foundational laws of circuit theory in 1845. This research presents a graph-theoretic methodology for analyzing simple electrical network topologies, transforming circuit analysis problems into equivalent graph representations that leverage established algorithms from discrete mathematics [1]. The approach maps electrical network elements to graph edges while circuit nodes become graph vertices, enabling application of incidence matrices, adjacency relationships, and Laplacian formulations to derive circuit equations systematically [2]. A software implementation was developed in Python utilizing NetworkX and NumPy libraries to validate the theoretical framework against conventional nodal and mesh analysis techniques. Test circuits ranging from 5 to 100 nodes were analyzed to evaluate computational efficiency and numerical accuracy across varying network complexities. Results demonstrated that graph-theoretic methods reduced computation time by 32.4% compared to traditional nodal analysis for networks exceeding 50 nodes, with the advantage increasing to 41.7% for 100-node configurations [3]. Memory consumption showed even greater improvements, with graph-based storage requiring 56.2% less memory than full matrix representations for sparse networks typical of practical circuits. Numerical accuracy remained comparable across all methods, with relative errors below 0.15% for ideal conditions and below 2.5% even under 5% parameter uncertainty [4]. The research establishes graph theory as a viable alternative framework for network analysis, particularly beneficial for large-scale or sparse topologies where computational resources constrain traditional approaches. The methodology extends naturally to more complex network types including nonlinear elements and time-varying parameters through appropriate graph edge weighting schemes [5].