MATLAB Implementation of Graph Theory Algorithms with Code Examples

In this article, we provide a comprehensive exploration of graph theory algorithms implemented in MATLAB. Graph theory algorithms are computational methods within computer science that solve problems related to graphs, which consist of nodes (vertices) and the edges connecting them. These algorithms address fundamental problems including shortest path finding, minimum spanning tree construction, maximum flow computation, and graph traversal techniques.

First, we introduce shortest path algorithms, which determine the minimum distance between nodes in a graph. These can find the shortest path between two specific nodes or compute distances from one node to all others. We will detail popular algorithms like Dijkstra's algorithm (using priority queues for greedy node selection) and Bellman-Ford algorithm (handling negative weights through relaxation iterations), including MATLAB implementation approaches using adjacency matrices and distance vector updates.

Next, we explore minimum spanning tree algorithms that construct trees connecting all nodes with minimal total edge cost. These have widespread applications in network design and logistics planning. We will examine Prim's algorithm (building trees by adding minimum-weight edges from connected components) and Kruskal's algorithm (using union-find data structures to merge disjoint sets), with MATLAB code examples demonstrating edge sorting and cycle detection techniques.

Then, we cover maximum flow algorithms that compute the maximum possible flow from source to sink nodes in flow networks. These are crucial for transportation planning and communication networks. We will analyze Ford-Fulkerson method (using residual graphs and augmenting paths) and Edmonds-Karp implementation (employing BFS for path finding), including MATLAB implementations with capacity matrices and flow accumulation tracking.

Finally, we discuss graph traversal algorithms that systematically visit all nodes by following graph paths. These are essential for search engines and social network analysis. We will detail depth-first search (DFS using stack-based recursion or iteration) and breadth-first search (BFS using queue-based level ordering), with MATLAB code examples showing vertex coloring techniques and adjacency list traversals.