different between graph vs quadrangulation

English

Etymology

Shortening of graphic formula. From 1878; verb from 1889.

Pronunciation

• (Received Pronunciation) IPA^(key): /????f/
• (US, Northern England) IPA^(key): /??æf/
• Rhymes: -??f, -æf

Noun

graph (plural graphs)

1. (applied mathematics, statistics) A data chart (graphical representation of data) intended to illustrate the relationship between a set (or sets) of numbers (quantities, measurements or indicative numbers) and a reference set, whose elements are indexed to those of the former set(s) and may or may not be numbers.

   Hyponyms: bar graph, line graph, pie graph
2. (mathematics) A set of points constituting a graphical representation of a real function; (formally) a set of tuples ${\displaystyle (x_{1},x_{2},\ldots ,x_{m},y)\in \mathbb {R} ^{m+1}}$, where ${\displaystyle y=f(x_{1},x_{2},\ldots ,x_{m})}$ for a given function ${\displaystyle f:\mathbb {R} ^{m}\rightarrow \mathbb {R} }$. See also Graph of a function on Wikipedia.Wikipedia
   • 1969 [MIT Press], Thomas Walsh, Randell Magee (translators), I. M. Gelfand, E. G. Glagoleva, E. E. Shnol, Functions and Graphs, 2002, Dover, page 19,

     Let us take any point of the first graph, for example, ${\displaystyle \textstyle x={\frac {1}{2}},y={\frac {4}{5}}}$, that is, the point ${\displaystyle \textstyle M_{1}({\frac {1}{2}},{\frac {4}{5}})}$.

3. (graph theory) A set of vertices (or nodes) connected together by edges; (formally) an ordered pair of sets ${\displaystyle (V,E)}$, where the elements of ${\displaystyle V}$ are called vertices or nodes and ${\displaystyle E}$ is a set of pairs (called edges) of elements of ${\displaystyle V}$. See also Graph (discrete mathematics) on Wikipedia.Wikipedia

   Hyponyms: directed graph, undirected graph, tree

   • 1973, Edward Minieka (translator), Claude Berge, Graphs and Hypergraphs, Elsevier (North-Holland), [1970, Claude Berge, Graphes et Hypergraphes], page vii,

     Problems involving graphs first appeared in the mathematical folklore as puzzles (e.g. Königsberg bridge problem). Later, graphs appeared in electrical engineering (Kirchhof's Law), chemistry, psychology and economics before becoming a unified field of study.

   • 1997, Fan R. K. Chung, Spectral Graph Theory, American Mathematical Society, page 1,

     Spectral graph theory has a long history. In the early days, matrix theory and linear algebra were used to analyze adjacency matrices of graphs. Algebraic methods are especially effective in treating graphs which are regular and symmetric.

4. (topology) A topological space which represents some graph (ordered pair of sets) and which is constructed by representing the vertices as points and the edges as copies of the real interval [0,1] (where, for any given edge, 0 and 1 are identified with the points representing the two vertices) and equipping the result with a particular topology called the graph topology.

   Synonym: topological graph

   • 2008, Unnamed translators (AMS), A. V. Alexeevski, S. M. Natanzon, Hurwitz Numbers for Regular Coverings of Surfaces by Seamed Surfaces and Cardy-Frobenius Algebras of Finite Groups, V. M. Buchstaber, I. M. Krichever (editors), Geometry, Topology, and Mathematical Physics: S.P. Novikov's Seminar, 2006-2007, American Mathematical Society, page 6,

     First, let us define its 1-dimensional analog, that is, a topological graph. A graph ${\displaystyle \Delta }$ is a 1-dimensional stratified topological space with finitely many 0-strata (vertices) and finitely many 1-strata (edges). […] A graph such that any vertex belongs to at least two half-edges we call an s-graph. Clearly the boundary ${\displaystyle \partial \Omega }$ of a surface ${\displaystyle \Omega }$ with marked points is an s-graph.

     A morphism of graphs ${\displaystyle \varphi :\Delta '\rightarrow \Delta ''}$ is a continuous epimorphic map of graphs compatible with the stratification; i.e., the restriction of ${\displaystyle \varphi }$ to any open 1-stratum (interior of an edge) of ${\displaystyle \Delta '}$ is a local (therefore, global) homeomorphism with appropriate open 1-stratum of ${\displaystyle \Delta ''}$.

5. (category theory, of a morphism f) A morphism ${\displaystyle \Gamma _{f}}$ from the domain of ${\displaystyle f}$ to the product of the domain and codomain of ${\displaystyle f}$, such that the first projection applied to ${\displaystyle \Gamma _{f}}$ equals the identity of the domain, and the second projection applied to ${\displaystyle \Gamma _{f}}$ is equal to ${\displaystyle f}$.
6. (linguistics, typography) A graphical unit on the token-level, the abstracted fundamental shape of a character or letter as distinct from its ductus (realization in a particular typeface or handwriting on the instance-level) and as distinct by a grapheme on the type-level by not fundamentally distinguishing meaning.

   Synonym: glyph

Usage notes

• In mathematics, the graphical representation of a function sense is generally of interest only at an elementary level.
  • Nevertheless, the term vertex-edge graph is sometimes used in educational texts to distinguish the graph theory sense.
• (points constituting a graphical representation of a function):
  • A graph is similar to, but not the same as a (real) function (as defined formally).
    • The function ${\displaystyle f}$ is a set of ordered pairs ${\displaystyle (x,f(x))}$, where ${\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})}$ is a point in ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle f(x)}$ is a point in ${\displaystyle \mathbb {R} }$.
    • A graph of ${\displaystyle f}$ is a set of points (represented as n-tuples) ${\displaystyle (x_{1},x_{2},\ldots ,x_{n},f(x_{1},x_{2},\ldots ,x_{n}))\in \mathbb {R} ^{n+1}}$.
• (graph theory):
  • A graph ${\displaystyle G=(V,E)}$ may be defined such that the elements of ${\displaystyle E}$ are ordered pairs or unordered pairs.
    • If the pairs are unordered, ${\displaystyle G}$ may be called an undirected graph and the elements of ${\displaystyle E}$ are called edges.
    • If the pairs are ordered, ${\displaystyle G}$ is called a directed graph or digraph and the elements of ${\displaystyle E}$ may be called arcs; the notation ${\displaystyle G=(V,A)}$ is sometimes used.
    • If the two vertices of an edge represent the same point, the edge may be called a loop.

Hyponyms

Derived terms

Related terms

Translations

Verb

graph (third-person singular simple present graphs, present participle graphing, simple past and past participle graphed)

1. (transitive) To draw a graph.
2. (transitive, mathematics) To draw a graph of a function.

Synonyms

• (draw a graph of): chart, plot

Translations

See also

• axis
• Cartesian product
• chart
• diagram
• grapheme
• plot

References

Further reading

• Chart on Wikipedia.Wikipedia
• Graph of a function on Wikipedia.Wikipedia
• Graph theory on Wikipedia.Wikipedia
• Graph (discrete mathematics) on Wikipedia.Wikipedia
• Graph (topology) on Wikipedia.Wikipedia
• Graph (abstract data type) on Wikipedia.Wikipedia
• Conceptual graph on Wikipedia.Wikipedia
• Glyph (disambiguation) on Wikipedia.Wikipedia
• Tree (disambiguation) on Wikipedia.Wikipedia
• Graph on Encyclopedia of Mathematics
• graph on nLab
• graph of a function on nLab
• graph of a functor on nLab
  • Function Graph on Wolfram MathWorld
• Graph on Wolfram MathWorld
  • Graph Vertex on Wolfram MathWorld
  • Graph Edge on Wolfram MathWorld

Anagrams

• phrag

English

Noun

quadrangulation (countable and uncountable, plural quadrangulations)

1. (mathematics) A graph embedded in a surface such that every face is a quadrangle.
2. The act or result of quadrangulating a surface.

Hypernyms

• polygonation

Coordinate terms

• triangulation