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When saying graph in mathematics, it can be either a graph of a function, or a graph in graph theory. However mathematically they have nothing in common. How did they get the same name?

I know graph theory has been invented by Euler (the seven bridges of Königsbergen, 1736), but I haven't been able to find the history of graphing a function. My guess is that Leibniz had done that as well while discovering functions.

Of course, they both have something related to graphical things, but you might as well say that about geometry, topology and often also combinatorics, so that isn't really a statistifactory argument.

How did 'graphing' a function and 'graph' theory get the same name? Was it a confusion, or did they have a common ancestor?

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  • $\begingroup$ If you look on the Wikipedia pages for "graph of a function" and "graph theory" and compare the tables of links to the same pages in other languages then you'll see that in most (but not all) languages besides English the term graph for a function's graph and for the combinatorial object are not the same. $\endgroup$ – KCd Jul 10 '15 at 19:04
  • $\begingroup$ @KCd I'm not sure if I'd agree with that. Dutch ('grafiek' vs. 'graaf'), French ('graphe' vs. 'graphe d'une function'), and German ('Graph' vs. 'Funktionsgraph'). Chinese (函数图形 vs.图, where 函数 means function), which is You see, the words have just the word for function added. $\endgroup$ – wythagoras Jul 10 '15 at 19:12
  • $\begingroup$ I was not counting the examples with the word function in them. Though why do you single out Dutch as an example of that? I do not know Dutch, but grafiek and graaf do not involve the word for function (functie). $\endgroup$ – KCd Jul 10 '15 at 19:46
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    $\begingroup$ For history of function graphs see hsm.stackexchange.com/questions/2378/… and dam.brown.edu/people/mumford/beyond/coursenotes/2006PartIb.pdf $\endgroup$ – Conifold Jul 11 '15 at 0:58
  • $\begingroup$ @KCd No, that was just an example of the similarity between them. $\endgroup$ – wythagoras Jul 18 '15 at 19:19
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All these words come from the same Greek root, "graphein" means to write in Greek. Later the meaning extended to include drawing. Also graphite, graphology, graphics (drawing), graphoman etc. Also photography, biography, histogram, cardiogram, grammar and telegraph have the same root. Graphs of functions and graphs in combinatorics are something that we draw, pictures.

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  • $\begingroup$ γράφω (“to scratch, to scrape, to graze”) and γραμματικὴ (art) are different roots. $\endgroup$ – Conifold Jul 11 '15 at 0:53
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    $\begingroup$ @Conifold. On the contrary: γράφω and γράμμα (<*graph-ma) are from the same root. $\endgroup$ – fdb Jul 11 '15 at 12:17
  • $\begingroup$ @fdb Interesting, so φω and μα are suffixes. What is the original meaning of γρά? $\endgroup$ – Conifold Jul 13 '15 at 18:40
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    $\begingroup$ @Conifold. No, the underlying Indo-European root is *gerbh-, zero-grade *grbh-, ‘to scratch’. In Greek the latter develops regularly to graph-, whence the verb (present) graph-ō, (aorist) e-grap-s-a, etc. -mat- is a common formative for neuter nouns, so graph+mat- > (nominative) gramma, (genitive) grammat-os, adjective grammat-ik-os, etc. $\endgroup$ – fdb Jul 13 '15 at 19:30

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