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I am currently working on an assignment for my discrete mathematics lecture. Here we are currently looking at triangulations, that is, graphs which have the following attributes: G is a graph with minimum degree of 3. G is a triangulation if every area in the graph is surrounded by exactly 3 edges.

Now, let vi be the number of vertices of degree 3. Show that the following is true:

3 * v3 + 2 * v4 + v5 = v7 + 2 * v8 + 3 * v9 + ... + (X - 6) vX + 12

Whereas X is the maximum degree in G.

Now, I was wondering whether this notation is at all exact. I understand what is implied by the "..." in there and see how the whole thing is supposed to work, but is it clear enough that you could, for example, say that when X = 4, you don't even include the (X - 6)vX part? Because if you do, this is easily shown to be wrong. If you don't, it probably is true. But is that something that is to be considered given through this notation? Or can I just give an example where it isn't true? I am simply not sure whether this is in any ways precise, when alternatively it would be easily written as a sum instead.

Thank you very much for reading!

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    $\begingroup$ This is just a device that will spare you the need to read into the more cumbersome $\Sigma_{k=7}^X (X-6)v_ k$. As long as it is clear what it means, I am fine with that and indeed appreciate not having to examine the $\Sigma$. $\endgroup$ – Sam Skywalker Jun 11 '19 at 15:08
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    $\begingroup$ As for if the index is smaller than what the example would have you believe... If I were to define, say, $[n] = \{1,2,3,\dots,n\}$ then it should be clear from context that by $[2]$ I mean $\{1,2\}$. Could I have been more precise and written it instead as $[n]=\{x\in\Bbb N^+~:~1\leq x\leq n\}$? Sure. Should it bother you that a $3$ appeared in the definition which doesn't appear when we talk about $[2]$? Perhaps, but it is largely ignored since the notation using "$\dots$" is natural and useful. $\endgroup$ – JMoravitz Jun 11 '19 at 15:12
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You can use $\ldots$ as long as it is clear what is meant to be filled in there; you don’t always have to write everything out or change notation, as long as the meaning of the dots is obvious.

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  • $\begingroup$ But is it as precise? I feel like it is open for interpretation and couldn't I interpret it as something that causes it to be just wrong? The lecture is always rather specific on our notation and demands it to be as precise as possible. With such a notation, I feel like it is too imprecise and I could just choose to interpret it in a ways that would cause it to be wrong. Would that be wrong? $\endgroup$ – Alan Jones Jun 11 '19 at 15:13

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