Graph Theory: Mathematical Tricks and Techniques

Graph theory, a branch of mathematics, provides a powerful framework for modelling and analyzing various real-world problems. While the visual representation of graphs is often helpful, there are numerous mathematical techniques and tricks that can simplify and enhance the problem-solving process.

Matrix Representation

• Adjacency Matrix: Represents the graph as a square matrix where rows and columns correspond to vertices, and entries indicate the presence or absence of an edge.
• Incidence Matrix: Represents the graph as a rectangular matrix where rows correspond to edges and columns correspond to vertices. Entries indicate the endpoints of each edge.

Graph Isomorphism

• Graph isomorphism: Two graphs are isomorphic if there exists a one-to-one correspondence between their vertices such that corresponding vertices have the same degree and corresponding edges connect the same pairs of vertices.
• Techniques: Check the number of vertices, edges, degrees, and vertex sequences to determine isomorphism.

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Eulerian Paths and Circuits

• Euler’s theorem: A connected graph has a Eulerian circuit if and only if all vertices have even degree.
• Fleury’s algorithm: A greedy algorithm for finding a Eulerian circuit in a connected graph with all vertices of even degree.

Hamiltonian Paths and Cycles

• Hamiltonian path: A path that visits every vertex exactly once.
• Hamiltonian cycle: A closed Hamiltonian path.
• Brute force search: A naive approach to find Hamiltonian paths and cycles, but it can be computationally expensive for large graphs.
• Heuristic algorithms: Approximate algorithms that provide good solutions but may not guarantee the optimal solution.

Shortest Paths

• Dijkstra’s algorithm: A greedy algorithm for finding the shortest paths from a single source vertex to all other vertices in a weighted graph.
• Bellman-Ford algorithm: An algorithm that can handle negative edge weights and detect negative cycles.
• Floyd-Warshall algorithm: An algorithm for finding the shortest paths between all pairs of vertices in a weighted graph.

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Minimum Spanning Trees

• Prim’s algorithm: A greedy algorithm that starts with a single vertex and gradually adds edges to the MST.
• Kruskal’s algorithm: A greedy algorithm that sorts edges by weight and adds them to the MST as long as they do not create a cycle.

Graph Colouring

• Graph colouring problem: Assign colours to vertices such that no adjacent vertices have the same colour.
• Greedy algorithms: Simple heuristics for graph colouring, such as the Welsh-Powell algorithm and the largest degree first algorithm.
• Exact algorithms: Branch-and-bound and backtracking algorithms for finding optimal colourings.

Network Flows

• Max-flow min-cut theorem: The maximum flow through a network is equal to the capacity of the minimum cut.
• Ford-Fulkerson algorithm: An algorithm for finding the maximum flow in a network.
• Edmonds-Karp algorithm: A variant of the Ford-Fulkerson algorithm that uses BFS to find augmenting paths.

Conclusion-

These are just a few of the many mathematical tricks and techniques that can be applied to graph theory problems. By mastering these techniques, you can effectively solve a wide range of challenges in various fields, from computer science and engineering to biology and social sciences.

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From
Ms. Priyavada
Assistant Professor
Department of Mathematics
Lingaya’s Vidyapeeth
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