graph

Etymology
Shortening of. From 1878; verb from 1889.

Noun

 * 1)  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.
 * 2)  A set of points constituting a graphical representation of a real function;  a set of tuples $$(x_1, x_2, \ldots, x_m, y)\in\R^{m+1}$$, where $$y=f(x_1, x_2, \ldots, x_m)$$ for a given function $$f: \R^m\rightarrow\R$$. See also
 * 3) * 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, $$\textstyle x=\frac 1 2, y=\frac 4 5$$, that is, the point $$\textstyle M_1(\frac 1 2,\frac 4 5)$$.
 * 1)  A set of vertices (or nodes) connected together by edges;  an ordered pair of sets $$(V,E)$$, where the elements of $$V$$ are called vertices or nodes and $$E$$ is a set of pairs (called edges) of elements of $$V$$. See also
 * 2) * 1973, Edward Minieka (translator),, 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.
 * 1)  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.
 * 2) * 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,, page 6,
 * First, let us define its 1-dimensional analog, that is, a topological graph. A graph $$\Delta$$ 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 $$\partial\Omega$$ of a surface $$\Omega$$ with marked points is an s-graph.
 * A morphism of graphs $$\varphi: \Delta'\rightarrow\Delta$$ is a continuous epimorphic map of graphs compatible with the stratification; i.e., the restriction of $$\varphi$$ to any open 1-stratum (interior of an edge) of $$\Delta'$$ is a local (therefore, global) homeomorphism with appropriate open 1-stratum of $$\Delta$$.
 * 1)  A morphism $$\Gamma_f$$ from the domain of $$f$$ to the product of the domain and codomain of $$f$$, such that the first projection applied to $$\Gamma_f$$ equals the identity of the domain, and the second projection applied to $$\Gamma_f$$ is equal to $$f$$.
 * 2)  A graphical unit on the, the abstracted fundamental shape of a character or letter as distinct from its ductus (realization in a particular typeface or handwriting on the ) and as distinct by a  on the  by not fundamentally distinguishing.
 * 1) * 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,, page 6,
 * First, let us define its 1-dimensional analog, that is, a topological graph. A graph $$\Delta$$ 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 $$\partial\Omega$$ of a surface $$\Omega$$ with marked points is an s-graph.
 * A morphism of graphs $$\varphi: \Delta'\rightarrow\Delta$$ is a continuous epimorphic map of graphs compatible with the stratification; i.e., the restriction of $$\varphi$$ to any open 1-stratum (interior of an edge) of $$\Delta'$$ is a local (therefore, global) homeomorphism with appropriate open 1-stratum of $$\Delta$$.
 * 1)  A morphism $$\Gamma_f$$ from the domain of $$f$$ to the product of the domain and codomain of $$f$$, such that the first projection applied to $$\Gamma_f$$ equals the identity of the domain, and the second projection applied to $$\Gamma_f$$ is equal to $$f$$.
 * 2)  A graphical unit on the, the abstracted fundamental shape of a character or letter as distinct from its ductus (realization in a particular typeface or handwriting on the ) and as distinct by a  on the  by not fundamentally distinguishing.

Usage notes

 * In mathematics, the graphical representation of a function sense is generally of interest only at an elementary level.
 * Nevertheless, the term is sometimes used in educational texts to distinguish the graph theory sense.
 * A graph is similar to, but not the same as a (real) function (as defined formally).
 * The function $$f$$ is a set of ordered pairs $$(x, f(x))$$, where $$x=(x_1, x_2, \ldots, x_n)$$ is a point in $$\R^n$$ and $$f(x)$$ is a point in $$\R$$.
 * A graph of $$f$$ is a set of points (represented as n-tuples) $$(x_1, x_2, \ldots, x_n, f(x_1, x_2, \ldots, x_n))\in\R^{n+1}$$.
 * A graph $$G=(V,E)$$ may be defined such that the elements of $$E$$ are ordered pairs or unordered pairs.
 * If the pairs are unordered, $$G$$ may be called an and the elements of $$E$$ are called.
 * If the pairs are ordered, $$G$$ is called a or  and the elements of $$E$$ may be called ; the notation $$G=(V,A)$$ is sometimes used.
 * If the two vertices of an edge represent the same point, the edge may be called a.
 * If the pairs are ordered, $$G$$ is called a or  and the elements of $$E$$ may be called ; the notation $$G=(V,A)$$ is sometimes used.
 * If the two vertices of an edge represent the same point, the edge may be called a.

Hyponyms

 * See also
 * See also Thesaurus:graph

Translations

 * Afrikaans: grafiek
 * Albanian:
 * Arabic: مُخَطَّط بَيَانِيّ
 * Armenian:
 * Asturian:
 * Azerbaijani: qrafik
 * Belarusian: гра́фік, дыягра́ма
 * Bengali: গ্রাফ
 * Bulgarian: ,
 * Burmese:
 * Catalan:
 * Chinese:
 * Mandarin:
 * Czech:
 * Danish:
 * Dutch:
 * Esperanto: grafeo
 * Estonian: graafik
 * Finnish:, kaaviokuva
 * French:
 * Galician:
 * Georgian: გრაფიკი
 * German:, Funktionsgraph
 * Greek:
 * Hebrew:
 * Hindi: लेखाचित्र, ग्राफ़
 * Hungarian: ,
 * Indonesian:
 * Irish: graf
 * Italian:
 * Japanese: ,
 * Kazakh: график
 * Khmer: ពិន្ទុរេខីយ
 * Korean:, 도표(圖表)
 * Kyrgyz:
 * Lao: ເສັ້ນສະແດງ, ກຣາຟ
 * Latvian: grafs, grafiks
 * Lithuanian: grafas, grafikas
 * Macedonian: графико́н
 * Malay:
 * Maori: kauwhata
 * Mongolian:
 * Cyrillic:
 * Mongolian: ᠭᠷᠠᠹᠢᠻ
 * Norwegian:
 * Bokmål:
 * Persian: ,
 * Polish:
 * Portuguese:
 * Romanian:
 * Russian: ,
 * Scottish Gaelic: graf
 * Serbo-Croatian:
 * Cyrillic: гра̏ф
 * Roman:
 * Slovak:
 * Slovene: graf
 * Spanish:
 * Swahili:
 * Swedish: ,
 * Tagalog: talangguhit
 * Tajik:
 * Thai:
 * Turkish:
 * Turkmen: grafik
 * Ukrainian: гра́фік, діагра́ма
 * Urdu: گراف
 * Uzbek:
 * Vietnamese:
 * Welsh: graff


 * Belarusian: граф
 * Bulgarian: граф
 * Catalan:
 * Chinese:
 * Mandarin: ,
 * Czech:
 * Danish:
 * Dutch:
 * Finnish: ,
 * French:
 * German:
 * Greek: ,
 * Hebrew:
 * Hindi: लेखाचित्र
 * Hungarian:
 * Irish: graf
 * Italian:
 * Japanese: ,
 * Korean:, 도표(圖表)
 * Polish:
 * Portuguese:
 * Romanian:
 * Russian:
 * Serbo-Croatian:
 * Cyrillic: гра̏ф
 * Roman:
 * Slovak:
 * Spanish:
 * Swedish:
 * Thai:
 * Ukrainian: граф
 * Vietnamese:

Verb

 * 1)  To draw a graph, to record graphically.
 * 2)  To draw a graph of a function.
 * 1)  To draw a graph of a function.

Translations

 * Galician: debuxar un gráfico de
 * Irish: graf
 * Italian: graficare
 * Spanish:
 * Swedish:, rita diagram