Graph Theory Algorithms with MATLAB Implementations

This article presents three fundamental graph theory algorithms along with their MATLAB implementations, designed to solve the following problems: 1. Warshall-Floyd algorithm for finding shortest paths between any two vertices in weighted graph G = (V, E, F), and Kruskal's cycle avoidance method. - The Warshall-Floyd algorithm computes shortest distances between all vertex pairs using dynamic programming. It iteratively improves distance estimates by considering intermediate vertices, implementing triple nested loops with time complexity O(n³). - Kruskal's algorithm constructs minimum spanning trees by sorting edges by weight and adding them in ascending order while checking for cycles using union-find data structures, ensuring no cycles are formed during tree construction. 2. Hungarian algorithm for finding maximum matching in bipartite graphs and the feasible labeling method for optimal matching. - The Hungarian algorithm systematically explores augmenting paths to find maximum bipartite matching. Implementation involves maintaining match arrays and using DFS/BFS to find alternating paths that improve current matching. - The feasible labeling algorithm uses vertex labeling and slack variables to find optimal matches through iterative label updates and matching adjustments, employing greedy strategies for efficient bipartite matching. 3. Ford-Fulkerson labeling algorithm for computing maximum flow starting from a feasible flow f. - This classic algorithm finds augmenting paths in residual networks and updates flow values until no improvement is possible. Implementation uses labeling techniques to track path discoveries and capacity constraints, with Edmonds-Karp variant ensuring polynomial time complexity. These algorithms and their MATLAB implementations provide robust solutions for various graph theory problems, featuring detailed code comments and optimization techniques for practical applications. The content is structured to facilitate understanding and immediate implementation for researchers and practitioners.