MATLAB Implementation of Graph Theory Toolbox

Implementing a graph theory toolbox in MATLAB can cover various classical algorithms widely applied in network analysis, transportation planning, social networks, and other domains. While MATLAB includes built-in graph theory functions, users can extend functionality through custom implementations.

Graph Representation MATLAB supports two primary representation methods: adjacency matrices and edge lists. Adjacency matrices are suitable for dense graphs, while edge lists better handle sparse graphs. Built-in functions `graph` and `digraph` create undirected and directed graphs respectively. For code implementation, users can initialize graphs using syntax like G = graph(A) where A is an adjacency matrix.

Shortest Path Algorithms Common algorithms include Dijkstra's algorithm and Floyd-Warshall algorithm. Dijkstra works for graphs with non-negative weights, while Floyd-Warshall handles graphs with negative weights (without negative cycles). MATLAB's `shortestpath` function encapsulates these algorithms - users simply input the adjacency matrix and source/target nodes. Implementation example: [path,distance] = shortestpath(G,s,t).

Search Algorithms Depth-First Search (DFS) and Breadth-First Search (BFS) form the foundation for graph traversal, used for connectivity detection and path finding. These can be implemented in MATLAB using recursion (for DFS) or queue structures (for BFS). Code structure typically involves maintaining visited node lists and stack/queue operations.

Matching Problems Maximum matching in bipartite graphs can be solved using the Hungarian algorithm. Although MATLAB doesn't provide this algorithm directly, it can be implemented through linear programming (using `linprog`) or manual coding using augmentation path techniques.

Max-Flow Problems Ford-Fulkerson and Edmonds-Karp algorithms are classical methods for max-flow problems. MATLAB's `maxflow` function calculates maximum flow and minimum cut in graphs. The function uses residual networks and augmentation paths internally, accessible via [maxFlow,flowMatrix] = maxflow(G,s,t).

K-Core Computation K-core represents an important subgraph concept for network structure analysis. It's solved by iteratively removing nodes with degree less than k. In MATLAB, this can be implemented using while-loops with conditional degree checks, updating the graph structure each iteration until convergence.

By combining these fundamental algorithms, users can build powerful graph theory toolboxes capable of handling various complex network analysis requirements.