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Discrete mathematics

Discrete mathematics, also called finite mathematics or decision mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the notion of continuity. Objects studied in finite mathematics are largely countable sets such as integers, finite graphs, and formal languages.

Discrete mathematics has become popular in recent decades because of its applications to computer science. Concepts and notations from discrete mathematics are useful to study or describe objects or problems in computer algorithms and programming languages. In some mathematics curricula, finite mathematics courses cover discrete mathematical concepts for business, while discrete mathematics courses emphasize concepts for computer science majors.

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A labeled graph on 6 vertices and 7 edges

Informally speaking, a graph is a set of objects called points, nodes, or vertices connected by links called lines or edges. In a proper graph, which is by default undirected, a line from point A to point B is considered to be the same thing as a line from point B to point A. In a digraph, short for directed graph, the two directions are counted as being distinct arcs or directed edges. Typically, a graph is depicted in diagrammatic form as a set of dots (for the points, vertices, or nodes), joined by curves (for the lines or edges). Graphs have applications in both mathematics and computer science, and form the basic object of study in graph theory.

Applications of graph theory are generally concerned with labeled graphs and various specializations of these. Many problems of practical interest can be represented by graphs. The link structure of a website could be represented by a directed graph: the vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B. A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. For example if a graph represents a road network, the weights could represent the length of each road. A digraph with weighted edges in the context of graph theory is called a network. Networks have many uses in the practical side of graph theory, network analysis (for example, to model and analyze traffic networks).

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A Penrose tiling, an example of a tiling that can completely cover an infinite plane, but only in a pattern which is non-repeating (aperiodic).

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Image credit: xJaM

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...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with n+1 factors?

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Topics in Discrete mathematics

Major areas	Combinatorics	Graph Theory	Game theory
Automata theory Combinatorics Combinatorial topology Computational geometry Digital geometry Discrete geometry Graph theory Information theory Optimization Order theory Game theory	Combinatorics Combination Permutation Enumerative combinatorics Pascal's triangle Combinatorial proof Bijective proof List of combinatorics topics	Graph theory Glossary of graph theory Graph coloring Network Network theory List of network theory topics Hypergraph List of graph theory topics	Game theory Extensive form game Normal form game Non-cooperative, Cooperative games Symmetric game Zero-sum Sequential game Perfect information Determinacy Strategy Solution concept Nash equilibrium Backward induction Glossary of game theory List of games in game theory

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