Essential Graph Theory Implementations Using MATLAB Code

As a powerful numerical computing tool, MATLAB provides comprehensive graph theory functions that efficiently handle various graph-related problems. Whether for network analysis, path planning, or graph visualization, MATLAB offers complete solutions with robust algorithmic implementations.

Basic Graph Operations In MATLAB, undirected and directed graphs are created using the `graph` or `digraph` functions respectively. Vertices and edges can be defined through adjacency matrices or edge lists. The `plot` function enables direct graph visualization, facilitating intuitive analysis of graph structures with customizable node positioning and edge styling.

Shortest Path Algorithms The `shortestpath` function implements Dijkstra's algorithm, suitable for single-source shortest path problems without negative edge weights. For scenarios with negative edge weights, the `bellmanford` algorithm handles more general single-source shortest path cases. All-pairs shortest paths are computed using the `distances` function, which employs the Floyd-Warshall algorithm with O(n³) time complexity for complete graph analysis.

Graph Traversal Depth-First Search (DFS) and Breadth-First Search (BFS) are implemented through `dfsearch` and `bfsearch` functions, used for node traversal or connected component identification. The `conncomp` function detects graph connectivity and identifies strongly or weakly connected components, returning component indices for further analysis.

Minimum Spanning Tree The `minspantree` function generates minimum weight spanning trees using Prim's or Kruskal's algorithms, applicable to network design optimization problems with efficient greedy algorithm implementations.

Other Practical Functions `maxflow` computes network maximum flow supporting Ford-Fulkerson algorithm implementations. `centrality` series functions (e.g., `degree`, `betweenness`) analyze node importance through various centrality metrics. `toposort` implements topological sorting for directed acyclic graphs (DAGs), commonly used in task scheduling applications with cycle detection capabilities.

MATLAB's graph theory tools leverage matrix computation advantages to rapidly process large-scale graph data, making them suitable for both engineering applications and scientific research with optimized built-in functions.