Graph Theory
Graph theory is a branch of mathematics concerned with the study of graphs. Graphs are mathematical structures used to model pairwise relations between objects, where:
- Vertices (or nodes) represent objects.
- Edges (or links) represent the relationships between these objects.
History
The origins of graph theory can be traced back to:
- The Königsberg Bridge Problem posed by Leonhard Euler in 1736, which sought to determine whether it was possible to walk through the city of Königsberg crossing each of its seven bridges exactly once. Euler's solution to this problem laid the foundations of graph theory.
- Further development came in the 19th century with contributions from mathematicians like Arthur Cayley, who used trees to enumerate chemical structures, and Gustav Kirchhoff, whose work on electrical networks also contributed to graph theory.
Key Concepts
Graph theory encompasses several fundamental concepts:
- Types of Graphs: Directed graphs, undirected graphs, weighted graphs, multigraphs, and bipartite graphs.
- Graph Properties: Connectivity, degree of vertices, path, cycle, and graph isomorphism.
- Graph Algorithms:
- Graph Theory Applications:
- Computer Science: network routing, search engines, social network analysis.
- Operations Research: flow optimization, scheduling.
- Biology: modeling metabolic pathways, phylogenetic trees.
Notable Results
- Four Color Theorem: Proved by Kenneth Appel and Wolfgang Haken in 1976, stating that any map can be colored with four colors in such a way that no two adjacent regions have the same color.
- Euler's Formula: For a connected planar graph with V vertices, E edges, and F faces, V - E + F = 2.
- Kuratowski's Theorem: Characterizes planar graphs through forbidden minors.
External Links
Related Topics