knowledge.deck

Graph Theory

Graph Theory is a branch of mathematics that studies the properties of graphs, which are mathematical structures used to model pairwise relations between objects. This area is foundational in computer science and combinatorics, with applications in network analysis, biology, linguistics, and social sciences.
Sub-categories:

Covers fundamental graph theory concepts such as vertices, edges, paths, and cycles, forming the foundation for understanding graph properties and operations.

Focuses on graphs that can be drawn on a plane without any edges crossing, including their characterization through Kuratowski's and Wagner's theorems.

Deals with the assignment of colors to vertices or edges under certain constraints, a key concept in scheduling and map coloring problems.

Pertains to algorithms for solving graph-related problems, such as shortest path, maximum flow, and graph traversal algorithms.

Applies graph theoretical principles to the analysis of networks, including social networks, transportation networks, and the Internet.

Studies a type of acyclic graph that has a hierarchical structure, important for data organization and network theory.

Examines how vertices are connected in a graph, covering aspects such as connected components, cuts, and connectivity measures.

Focuses on the matching of graph vertices or edges in pairs, with applications in job assignments, network design, and marriage problems.

Looks at changing graphs over time, which is useful for modeling dynamic systems such as traffic networks or social media.

Concerns the study of graphs that are generated by some random process, with applications in network science and probabilistic method.

Deals with the deeper properties and characterizations of graph classes through invariants and graph morphisms.

Explores the relationship between graph theory and topology, including embeddings and genus.

Studies the properties of graph minors, which are graphs obtained by edge contraction, with applications in the graph structure theorem.

Investigates the spectrum of the adjacency matrix of graphs and its implications for graph properties.

Focuses on understanding the maximum or minimum size of a graph property within certain constraints.

Considers the counting of graphs that meet specific criteria and the implications this has for combinatorics and statistics.