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I've been writing some things in graph theory, and because of my lack of experience writing in mathematical notation I find it difficult to think of how to write something. I want to write (for regular graphs) "The degree of every vertex in set V is the same". I thought that maybe it would be (having the degree of a vertex be $g(v)$: $$\forall v_i, v_j \in V : g(v_i) = g(v_j)$$ The part I have a problem with is just the middle, where I'm tempted to write the "the following condition is true for the previous group" part as $\longrightarrow$(which I don't know if it would be correct), but even with the colon it doesn't seem correct to me, though I'm not sure why

Edit: I've just seen while reading about graph theory that the case I needed solved is just saying that the maximum degree and minimum degree of the set are equal, but my question about notation still stands. If I were to write it like that, what would the correct way be? Is it possible as I formulated it, or does it have to be changed in other part besides the colon in the middle?

Edit2: Some have asked why I find a need to formalize(not sure that'd be the word) the statement. I'm a student(not in university, still have a year left before that) and I have an assignment in which it would be better to be as formal as possible. Not sure if other countries' education systems have a similar thing, but mine has a whole subject(it's valued as any other class, such as math, physics, English...) in which we have to do a low-level research. It could be anyhitng, and I chose math, so it's important for me to write this the best way possible, and I thought this'd be the way. As I'm on vacation right now, I couldn't ask my teacher

TexDuck
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    "The degree of every vertex in set V is the same". --- It would help to explain why you feel that you need to restate this straightforward statement in mathy symbolic form. For instance, are you trying to analyze something about its logical structure, such as obtain an upper bound on the minimum number of quantifiers involved when it is expressed in first order logic? FYI, I'd probably write the statement as: "All vertices in V have the same degree." – Dave L. Renfro Sep 10 '24 at 11:37
  • Alternatively, $\exists d:\forall v\in V:g(v)=d$ or even $\forall v\in V:g(v)=d$ if the common degree is known. –  Sep 10 '24 at 11:47
  • @DaveL.Renfro Please post your "use words" comment as an answer, even though the OP has accepted another. If "as formal as possible" in the question really calls for logical notation the assignment is flawed. – Ethan Bolker Sep 10 '24 at 12:45
  • @Ethan Bolker: Just saw your comment. In an answer (with the assistance of google and knowing that I've used the term "symbolic clutter") I've included some other stuff along with my above comment. – Dave L. Renfro Sep 10 '24 at 14:09

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(1st paragraph originally a comment)

It would help to explain why you feel that you need to restate this straightforward statement in mathy symbolic form. For instance, are you trying to analyze something about its logical structure, such as obtain an upper bound on the minimum number of quantifiers involved when it is expressed in first order logic? FYI, I'd probably write the statement as: "All vertices in $V$ have the same degree."

Regarding your addition of "it would be better to be as formal as possible", I encourage you to read the answer to the mathoverflow question How can I improve my formal definitions?, especially the paragraph beginning with First, I avoided using too many symbols and all comments to the answer. For example, there I wrote the following in a comment: "Nearly 50 years ago (1970), Paul Halmos wrote "The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.".

See also the last paragraph of this MSE answer, my comments to this MSE answer, and my answer to the MSE question How formal or informal should math texts (written for different purposes) be?.

Dave L. Renfro
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The colon is perfectly fine. Some books would instead write it as $$g(v_i)=g(v_j)\ \forall v_i,v_j\in V$$ which may look more "normal" to you. It doesn't really matter as long as it is clear what is meant. One might also want to drop the unnecessary indices and then write $$g(v)=g(w)\ \forall v,w\in V$$ because if you are putting indices, you must have some sort of labelling anyway, so maybe it is more natural to write $$g(v_i)=g(v_j)\ \forall i,j\in I$$ if your vertex set is $\{v_i\}_{i\in I}$.

Hope this helps. :)

ultralegend5385
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I've seen a couple of ways in which you can represent statements of the form above.

Statement: "The degree of every vertex in the set V is the same"

  1. $\forall v_i, v_j\in V(g(v_i)=g(v_j))$
  2. $g(v_i) = g(v_j), \ \forall v_i, v_j \in V$

In style 1, you specify where the elements come from, and then the condition they satisfy. I've seen this style used in logic/proof-based books.

In style 2, you write the condition first, and then you specify the set from where the elements are picked. This style seems to be more common across different texts (apart from logic/proof books)

shgwani2804
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