Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Graph |
GraphNoun1. A drawing illustrating the relations between certain quantities plotted with reference to a set of axes. Verb1. Represent by means of a graph. 2. Plot upon a graph. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "graph" was first used: 1878. (references) |
| Domain | Definition |
Computing | Graph 1. |
Aerospace | A diagram indicating the relationship between two or more variables. (references) |
Information | Type of data structure used to represent the different items of a file and their synthetic relations. As in tree structures, the nodes are generally used to represent the entities ( items ) and the labelled branches or arrows between the nodes represent various type of relations between the first and the second ( terminating ) one. Source: European Union. (references) |
Language | The smallest discrete segment in a stretch of writing or print. Source: European Union. (references) |
Math | A set of items connected by edges. Each item is called a vertex or node. Formally, a graph is a set of vertices and a relation between vertices, adjacency. (references) |
Mathematics | A drawing depicting the relation between certain sets of numbers or quantities by means of a series of dots, lines, etc. , plotted with reference to a set of axes. Source: European Union. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
A graph is
- In linguistics, a letter or symbol such as an alphabetic letter, a Chinese character, or a hieroglyph.
- In mathematics, the word graph can have two meanings.
- In graph theory, a graph is an object consisting of vertices (or nodes) and edges (or arcs) between pairs of vertices.
- The graph of a function f : X -> Y is the set of all pairs (x,f(x))
- The graph of a relation, a generalisation of the graph of a function.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Graph."
(From Wikipedia, the free Encyclopedia)
This article is about graphs in graph theory. See graph of a function and graph of a relation for other uses of "graph" in mathematics.
In mathematics and computer science, a graph is a generalization of the simple concept of a set of dots, called vertices or nodes, connected by links, edges or arcs. "Nodes" and "arcs" are old notation. Depending on the applications, edges may or may not have a direction; edges joining a vertex to itself may or may not be allowed, and vertices and/or edges may be assigned weights, i.e. numbers. If the edges have a direction associated with them (indicated by an arrow in the graphical representation) we have a directed graph.
Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be formulated as questions about certain graphs. For example, the link structure of Wikipedia could be represented by a directed graph: the vertices are the articles in Wikipedia, and there's a directed edge from article A to article B if and only if A contains a link to B. Directed graphs are also used to represent finite state machines. The development of algorithms to handle graphs is therefore of major interest in computer science: see graph algorithms.
To do: Add more pictures here.
A graph with 6 vertices and 7 edges.
History
See graph theory.
Basic Formal Definitions
Definitions in graph theory vary in the literature. Here are the conventions used in this encyclopedia.
A directed graph (also called digraph or quiver) consists of
An undirected graph (or graph for short) is given by
- a set V of vertices, and
- a set E of edges, and
- maps s, t : E → V, where s(e) is the source and t(e) is the target of the directed edge e.
In a weighted graph or digraph, an additional function E → R associates a value with each edge, which can be considered its "cost"; such graphs arise in optimal route problems such as the traveling salesman problem.
- a set V of vertices,
- a set E of edges,
- a function w : E → P(V) which associates to each edge a two- or one-element subset of V, interpreted as the endpoints of the edge.
Normally, the vertices of a graph by their nature are undistinguishable. (Of course, they may be distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges). Some branches of graph theory require to uniquely identify vertices. If each vertex is given a label, then the graph is said to be labelled graph. Consequently, graphs without labels on vertices are called unlabelled.
For more definitions, see Glossary of graph theory.
Examples
Formal definition: V={1,2,3,4,5,6}, E={e1,e2,e3,e4,e5,e6,e7} and the function w(e1)={1,2}, w(e2)={2,3}, w(e3)={1,5}, w(e4)={2,5}, w(e5)={3,4}, w(e6)={4,5}, w(e7)={4,6}.
Pictorial representation of graphs
Graphs are often represented pictorially as follows: draw a dot for every vertex, and for every edge draw an arc connecting its endpoints. If the graph is directed, indicate the endpoint of an edge by an arrow.
Note that this graphical representation (a layout) should not be confused with the graph itself (the abstract, non-graphical structure). Very different layouts can correspond to the same graph (see http://www.aisee.com/gallery/graph23.htm ). All that matters is which vertices are connected to which others by how many edges.
There are different approaches to graph layout, and these are considered under a branch of graph theory termed as graph drawing.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Graph (mathematics)."
(From Wikipedia, the free Encyclopedia)
Mathematically, the graph of a function is the collection of all pairs (x, f(x)) of the function. Graphing is sometimes referred to as curve sketching.
The graph of the function
is {(1,a), (2,d), (3,c)}.
The graph of the cubic polynomial on the real line
is {(x,x3-9x) : x is a real number}. If the set is plotted on a Cartesian plane, the result is
Therefore the graph of a function on real numbers is identical to the graphic representation of the function. For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis.
The concept of the graph of a function is generalised to the graph of a relation. Note that althrough a function is always identified with its graph, they are not the same because it will happen two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned above is a surjection if its codomain is the real numbers but it is not if its codomain is the complex field.
See also:
- concavity
- critical point
- derivative
- extrema
- slope
- solution point
- vertical translation
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Graph of a function."
(From Wikipedia, the free Encyclopedia)
Graph theory is the branch of mathematics that examines the properties of graphs.
A graph with 6 vertices and 7 edges. Informally, a graph is a set of objects called vertices or nodes connected by links called edges or arcs. Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges).
For more and formal definitions, see Glossary of graph theory and Graph (mathematics).
Depending on the applications, edges may or may not have a direction; edges joining a vertex to itself may or may not be allowed, and vertices and/or edges may be assigned weights, i.e. numbers. If the edges have a direction associated with them (indicated by an arrow in the graphical representation) we have a directed graph, or digraph.
Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be formulated as questions about certain graphs. For example, the link structure of Wikipedia could be represented by a directed graph: the vertices are the articles in Wikipedia, and there's a directed edge from article A to article B if and only if A contains a link to B. Directed graphs are also used to represent finite state machines. The development of algorithms to handle graphs is therefore of major interest in computer science.
History
Leonhard Euler's paper on Seven Bridges of Königsberg is considered to be the first result in graph theory. It is also regarded as one of the first topological results in geometry; that is, it does not depend on any measurements. This illustrates the deep connection between graph theory and topology.
Graph problems
- Coloring graphs: the four color theorem
- Route problems:
- Seven bridges of Königsberg
- Minimum spanning tree
- Shortest path problem
- Route inspection problem (also called the "Chinese Postman Problem")
- Traveling salesman problem
- Flows:
- Max flow min cut theorem
- reconstruction conjecture
- Isomorphism problems (Graph matching)
- Canonical Labeling
- subgraph isomorphism and monomorphisms
- Maximal common subgraph
Important algorithms
- Dijkstra's algorithm
- Kruskal's algorithm
- Nearest neighbour algorithm
- Prim's algorithm
Generalizations
In a hypergraph an edge can connect more than two vertices.
An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.
Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.
In model theory, a graph is just a structure. But in that case, there is no limitations on the number of edges: it can be any cardinal number.
Related areas of mathematics
- Ramsey theory
- Combinatorics
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Graph theory."
(From Wikipedia, the free Encyclopedia)
An image (from Latin imago) or picture is a visual reproduction of an object or a person, either by using optics (using a camera, mirror, refraction, telescope, microscope, etc.), or by artistic methods such as drawing or painting.See also:
In mathematics, an image is a value or set of values of a function. Specifically, let f be a function from the set X to the set Y. If a is an element of X, then its image under f is the value f(a). If A is a subset of X, then its image under f is defined to be f(A) := {f(a) : a in A}. Finally, the image of f itself is f(X), the same as the range of f. In religion, an image is an idol or icon. In philosophy, an image is a conception or idea. In Comics, Image is a publisher of such characters as Spawn. See Image Comics.
- Imaging
- Raster graphics
- Vector graphics
- Voyager Golden Record
- Disk image: a CR-ROM ISO image or floppy image. It´s a file that contains sector by sector what is in a CD-ROM or floppy.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Image."
Synonym: GraphSynonym: graphical record (n). (additional references) |
| Domain | Usage | |
Screenplays | In fact I made a graph (The Simpsons; writing credit: Artur Brauner; Paul Hengge) | |
Movie/TV Titles | Down to Earth Part III: The Island of Graph (1974) Polka Graph (1947) | |
Source: compiled by the editor from various references; see credits. | ||
| Domain | Title | ||
Books | |||
Periodicals |
| ||
Theater & Movies | |||
High Tech |
| ||
Source: compiled by the editor from various references; see credits. | |||
| Thumbnail | Description & Credit | Thumbnail | Description & Credit |
 | Pictured are two scientists conferring over a graph. They are in lab coats in an office setting. The new technology available to the scientists today plays an important role in providing them with needed detailed information. Credit: Linda Bartlett (photographer). |  | Line graph showing AIDS Cases by Exposure Category and Year of Report 1985-1996, United States. Credit: CDC. |
 | Bar graph showing AIDS cases by age and sex, reported 1981-1996, United States. Credit: CDC. |  | "Canary" (movie) by Travis Casper. The actual graph flys much faster than this slowed down sequence, and it breathes, too. |
 | Here is an 8D graph -- a 3D array of time-dependent 3D vectors, using color to show a pressure wave propagating through the moving vectors. |  | "Surreal Sphere to Sphere" by Lennart Agborn. Use the Scrollbar to vary A. The graph initially comes up blank; you must vary A to see it. |
 | "Animated Plane 1" (movie) by Rod Rodrigues. This graph shows how to use Time to create a movie of a plane slicing a surface. |  | "Airplane" by Staffan Björkenstam. This graph illustrates the Doppler effect. Use the scrollbar to vary A (time) and B (Mach speed). From inside DPGraph, click on Edit for more information. |
 | Figure 46 (cont.) Shaeffer and Budenberg recording manometer, mechanism above, recording graph below. The instrument is within an enclosed case which is acted upon by water pressure. An amplification mechanism transmits the displacement to a pen which records the corresponding depth on a gridded sheet. Credit: Sailing for Science - the NOAA Fleet Then and Now. |  | Figure 41. Apstein instrument and graph for determining the depth of horizontal ly towed nets. First noted by Carl Apstein in 1909, this instrument was used for many years to determine the depth of net at various towing speeds and amount of tension on the towing line. Credit: Sailing for Science - the NOAA Fleet Then and Now. |
Source: pictures compiled by the editor from various references; see picture credits. | |||
| Subject | Topic | Quote |
Health | Find your weight on the bottom of the graph. (references) | |
A line graph that records urinary bladder pressure at various volumes. (references) | ||
The heart's electrical signals cause a pen to draw lines across a strip of graph paper in the ECG machine. (references) | ||
Business | The corresponding increase in expected output is shown in the graph on the bottom. (references) | |
The bar graph below is based on imports of golf clubs and balls through traditional channels. (references) | ||
The following graph (*) states the most recent published data on registered cancer mortality in Argentina. (references) | ||
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| "Graph" is generally used as a noun (singular) -- approximately 99.80% of the time. "Graph" is used about 1,021 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Noun (singular) | 99.8% | 1,019 | 7,270 |
| Lexical Verb (base form) | 0.2% | 2 | 245,945 |
| Total | 100.00% | 1,021 | N/A |
Source: compiled by the editor from several corpora; see credits.
Expressions using "graph": bar graph ♦ bipartite graph ♦ cause effect graph ♦ circular graph ♦ complete graph ♦ connected graph ♦ De Bruijn graph ♦ directed acyclic graph ♦ directed graph ♦ exploded circular graph ♦ function Graph Language ♦ graph Algorithm and Software Package ♦ graph coloring ♦ graph colouring ♦ graph generation language ♦ graph paper ♦ graph plotter ♦ graph reduction ♦ graph rewriting system ♦ linear Graph Notation ♦ Moore graph ♦ programmed Graph REwriting Systems ♦ regular graph. Additional references. | |
| Hyphenated Usage | |
Beginning with "graph": graph-based, Graph-Oriented Object Database, graph-paper, graph-plotter, graph-search, graph-searching. | |
Ending with "graph": Idea-o-graph, moist-o-graph, pho-to-graph, photo-graph, three-graph, view-graph. | |
| Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Expression | Frequency per Day | Expression | Frequency per Day |
graph | 877 | excel graph | 30 |
graph paper | 850 | print graph paper | 28 |
2003 death epidemic graph sars | 219 | function and graph | 28 |
bar graph | 172 | graph misleading | 28 |
chart and graph | 124 | graph software | 27 |
line graph | 111 | supply and demand graph | 27 |
graph paper printable | 108 | stock market graph | 26 |
graph theory | 76 | 3d graph | 26 |
free graph paper | 69 | free graph paper printable | 25 |
create graph | 58 | stock graph | 24 |
pie graph | 53 | graph skin | 23 |
circle graph | 53 | type of graph | 23 |
make a graph | 43 | graph tech | 22 |
climate graph | 40 | graph expo | 22 |
graph picture | 36 | graph paper template | 22 |
math graph | 36 | population graph | 21 |
making graph | 35 | java graph | 21 |
mortgage rate graph | 35 | abortion graph | 20 |
ti graph link | 31 | free graph | 18 |
graph maker | 30 | polar graph | 17 |
graph paper online | 17 | ||
| Source: compiled by the editor from various references; see credits. | |||
| Language | Translations for "graph"; alternative meanings/domain in parentheses. | |
Albanian | grafik (bend, chart, curvature, curve, diagram, graphic, graphical), ndërtoj grafik, diagramë (chart, diagram). (various references) | |
Arabic | خط بياني (diagram), أداة تسجيل, رسم بياني (diagram, outline, schema, scheme, silhouette). (various references) | |
Bulgarian | чертая график, крива (curve, sweep), график (curve, plot), диаграма (chart, curve, diagram, pattern, plan, plot, schema, scheme, sheet, tabulation). (various references) | |
Chinese | 图表 (Diagrammatic, diagrammatical, graphic, graphics). (various references) | |
Czech | grafikon (flow diagram), graf (chart, diagram), křivka (curve, undulation), diagram (chart, diagram, schema). (various references) | |
Danish | graf (graph a non-empty but finite set of vertices(or nodes)together with a set of edges that join pairs of distinct vertices). (various references) | |
Dutch | diagram (chart, charts, diagram, flow chart). (various references) | |
Farsi | هجای کلمه , نمودار (Chart, Conspectus, Diagram, Schema), نمایش هندسی , نقشه هندسی , گرافیک , طرح خطی , اشکال مختلف یک حرف , بانمودارنشان دادن , باگرافیک وطرح خطی ثبت کردن . (various references) | |
Finnish | taulukko (index, table, tablet, tabulation), käyrä (bent, crooked, curve, curved, diagram), diagrammi (diagram). (various references) | |
French | graphique (graphic, graphical, graphics). (various references) | |
German | schaubild (chart, diagram, flip chart), kurvenblatt (hydrostatic curves, The variations of ship hydrostatic data with draught are shown by a set of curves.Extremely useful in the assessment of end draughts and the stability of a ship in various conditions of loading.The calculations for such curves are now normally made by a c), graph (graph a non-empty but finite set of vertices(or nodes)together with a set of edges that join pairs of distinct vertices), graphische Darstellung (arithmetic chart), diagramm (chart, diagram, figure). (various references) | |
Greek | γραφική παράσταση (chart, diagram), διάγραμμα (chart, diagram). (various references) | |
Hebrew | תרשים (chart, design, diagram, outline, plan, sketch). (various references) | |
Hungarian | grafikon (chart, curve, diagram). (various references) | |
Indonesian | grafik. (various references) | |
Italian | grafico (chart, graphic, graphic designer, graphical, printer). (various references) | |
Japanese Kanji | 図表 (chart, diagram), グラ"ア印刷 (graffiti, graphic, graphic arts, graphic design, graphic designer, graphic display, graphic equalizer, graphical, graphical interface, graphics, graph-matching, photogravure). (various references) | |
Japanese Katakana | ずひょう (chart, diagram), グラフ . (various references) | |
Korean | 도표 (Chart, Diagram, graphic, graphics). (various references) | |
Manx | graaf (chart, plot). (various references) | |
Norwegian | diagram. (various references) | |
Pig Latin | aphgray.(various references) | |
Portuguese | grfico (graphic, graphical), gráfico (chart, diagram, graphic, graphic arts, graphics, spread sheet), representao grfica, mapa (chart, map, survey), esquema (cadre, design, draft, draught, game, model, pattern, plan, schema, scheme), esboo (groundwork), diagrama (blueprint, diagram, figure). (various references) | |
Romanian | grafic (chart, diagram, diagrammatic, graphic, graphical, graphically, schedule). (various references) | |
Russian | график (diagram, graphic chart, timeline, timetable), граф (count, earl), диаграмма (card, chart, diagram, figure, pattern). (various references) | |
Serbo-Croatian | grafikon (chart, index), predstaviti grafičkim putem. (various references) | |
Spanish | gráfico (chart, diagram, graphic, graphical, pictorial, picturesque, plot, vivid), gráfica (graphics). (various references) | |
Swedish | graf (grave, tomb). (various references) | |
Turkish | grafik (chart, tableau), işaret (augury, badge, character, chop, clew, clue, cue, device, distinguishing mark, earmark, ensign, foretoken, gesture, glimpse, harbinger, hint, index, indication, indicator, landmark, logo, logotype, marker, note, pip, prognostic, prognostication, representation, sign, signal, signature, symbol, symptom, token, touch, trace), eğri (aslant, aslope, awry, bent, cambered, cockeyed, crooked, curve, oblique, out of the straight, sinuous, skew, skewed, slanting, sloping, slouching, slouchy, tortuous, trajectory, untrue, warped, wry), diyagram (diagram), çizelge (chart, exploded view, table). (various references) | |
Turkmen | grafa (r). (various references) | |
Ukrainian | графік (chart, diagram, timetable), діаграма (chart, delineation, diagram, figure). (various references) | |
Vietnamese | máy in thạch (chromograph). (various references) | |
| Source: compiled by the editor from various translation references. | ||
| Language | Period | Translations |
| Greek | 700 BCE-300 CE | graphikos. (various references) |
| Latin | 500 BCE-Modern | graphicus. (various references) |
| Source: compiled by the editor from various references. | ||
Derivations | |
Words beginning with "graph": graphed, grapheme, graphemes, graphemic, graphemically, graphemics, graphic, graphical, graphically, graphicness, graphicnesses, graphics, graphing, graphite, graphites, graphitic, graphitizable, graphitization, graphitizations, graphitize, graphitized, graphitizes, graphitizing, grapholect, grapholects, graphological, graphologies, graphologist, graphologists, graphology, graphs. (additional references) | |
Words ending with "graph": allograph, anemograph, astrophotograph, autograph, autoradiograph, barograph, bathythermograph, cardiograph, choreograph, chromatograph, chromolithograph, chronograph, cinematograph, coronagraph, coronograph, cryptograph, diagraph, digraph, electrocardiograph, electroencephalograph, electromyograph, electroretinograph, encephalograph, epigraph, ergograph, hectograph, heliograph, holograph, homograph, hygrograph, ideograph, isograph, kymograph, lithograph, logograph, macrophotograph, magnetograph, microbarograph, micrograph, microphotograph, microradiograph, mimeograph, monograph, myograph, nomograph, odograph, oleograph, oscillograph, pantograph, paragraph, phonocardiograph. (additional references) | |
Words containing "graph": agrapha, agraphia, agraphias, agraphic, allographic, allographs, anemographs, angiocardiographic, angiocardiographies, angiocardiography, angiographic, angiographies, angiography, antipornographic, antipornography, aortographic, aortographies, aortography, arteriographic, arteriographies, arteriography, astrophotographer, astrophotographers, astrophotographies, astrophotographs, astrophotography, autobiographer, autobiographers, autobiographic, autobiographical, autobiographically, autobiographies, autobiography, autographed, autographic, autographically, autographies, autographing, autographs, autography, autoradiographic, autoradiographies, autoradiographs, autoradiography, barographic, barographs, bathythermographs, bibliographer, bibliographers, bibliographic, bibliographical. (additional references) | |
| |
"Graph" is suggested in spellcheckers for the following: Brapho, craph, Gamph, Gapdh, Garah, garf, garp, geaf, gerraff, giraf, Gracht, gradh, Grah, Grakhov, grap, graphe, grapht, graphy, grapo, grapp, grath, greap, gref, grep, grepo, grp, grrap, gruh, grup, raph, rapho. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "graph" (pronounced gra"f) |
| 3 | -r a" f | Raff. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
| Words within the letters "a-g-h-p-r" | |
-1 letter: harp. | |
-2 letters: gap, gar, hag, hap, pah, par, rag, rah, rap. | |
-3 letters: ag, ah, ar, ha, pa. | |
| Words containing the letters "a-g-h-p-r" | |
+1 letter: graphs. | |
+2 letters: agrapha, digraph, graphed, graphic, harping. | |
+3 letters: agraphia, agraphic, diagraph, digraphs, epigraph, grapheme, graphics, graphing, graphite, handgrip, harpings, isograph, myograph, odograph, parching, phrasing, prophage, ralphing, sharping, subgraph, trigraph, upgather. | |
+4 letters: agraphias, allograph, autograph, barograph, biography, diagraphs, diaphragm, digraphic, epigraphs, epigraphy, epiphragm, ergograph, geography, grapeshot, graphemes, graphemic, graphical, graphites, graphitic, hampering, handgrips, holograph, homograph, hypergamy, ideograph, isographs, kymograph, lagomorph, logograph, monograph, myographs, nomograph, odographs, oleograph, orography, orphanage, orphaning, paragraph, phalanger, pharynges, phonogram, photogram, phrasings, polygraph, preaching, prophages, repechage, reshaping, scarphing, serigraph, straphang, straphung, subgraphs, telegraph, threaping, trigraphs, typograph, upgathers, xylograph, zigamorph. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
| 1. Definition 2. Synonyms 3. Crosswords 4. Usage: Modern | 5. Usage: Commercial 6. Images: Slideshow 7. Images: Photo Album 8. Quotations: Non-fiction | 9. Usage Frequency 10. Expressions 11. Expressions: Internet 12. Translations: Modern | 13. Translations: Ancient 14. Derivations 15. Rhymes 16. Anagrams | 17. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
