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Short version:

If you take n vertices and connect them together with lines, you'll have n−1 lines, why n−1? Other than the obvious visual proof, is there something that could be more, I don't know, substantial?

O - dot

O------------------------O------------------------O Three dots, two lines. $n$ -> $n-1$

Longer version:

Other than the obvious there's three dots, there are two lines - therefore $n-1$ is there some sort of a proof? For example, I can easily see how $n$ disconnected dots give $n/2$ lines, because there must be $m$ groups of 2 dots or $m$ lines.

I've been thinking about this a bit, if there's n dots, only the first line requires two unique dots and every subsequent one requires only 1 additional. But I still don't see the $n-1$ in that. Please assist me with this triviality, thanks!

Dots/vertices, however you want to name them.

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    Can you explain more about what exactly you are talking about? *I* can't tell.2011-11-21
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    If you take $n$ vertices and connect them together with lines, you'll have $n-1$ lines, why $n-1$? Other than the obvious visual proof, is there something that could be more, I don't know, substantial?2011-11-21
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    Explain that in *the body of the question*, so that people do not have to read through all comments.2011-11-21
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    @NKLost You'll also have to be more clear about how you are connecting the dots. In my mind, connecting 3 dots yields three lines, and 4 dots yield 6 lines. I also see ways in which 234987 dots yield one line.2011-11-21
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    @NKLost That said, once you sufficiently rigorize the question, induction will work.2011-11-21
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    Dot, then line, dot, then line, ..., dot, then line, dot. There's one more dot than line. Isn't that enough of a proof?2011-11-21
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    (A remark from @Pete Clark's [notes on induction](http://math.uga.edu/~pete/3200induction.pdf).) Some scholars have suggested that what is essentially an argument by mathematical induction appears in the later middle Platonic dialogue Parmenides, lines 149a7-c3. But this argument is of mostly historical and philosophical interest. The statement in question is, very roughly, that if $n$ objects are placed adjacent to another in a linear fashion, the number of points of contact between them is $n − 1$. [contd]2011-11-21
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    [contd] (Maybe. To quote the lead in the wikipedia article on the Parmenides: "It is widely considered to be one of the more, if not the most, challenging and enigmatic of Plato’s dialogues.") There is not much mathematics here! Nevertheless, for a thorough discussion of induction in the Parmenides the reader may consult [Ac00] and the references cited therein.2011-11-21
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    Ultimately it doesn't have to be "in a linear fashion" - this generalizes to [finite] trees. (basically, you get to attach the "then line, dot" anywhere instead of at the dot you added most recently)2011-11-21

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