[The] function is non-negative at all the vertices of the structure S and positive at some vertex...
That statement has a specific meaning, and it is a proper and precise way of stating that meaning:
The function is not negative at any vertex of S (that is, there is no vertex of S where the function is negative). In addition, there exists at least one vertex of S where the function is positive.
Note that zero is neither negative nor positive; to say that a function is non-negative means that its value is zero or positive. To say a function is positive means that it has a value greater than zero.
Most native English speakers will not notice any difference due to of being present or absent in the example sentence. If you are concerned about it, change all to every:
The function is non-negative at every vertex of the structure S and is positive at some vertex...
Edit: onomatomaniak commented,
I wonder, though, if “some vertex” means not at least one vertex, as you indicate, but rather [precisely] one vertex. If I wanted to indicate at least one, my inclination would be to write some vertices, not some vertex.
Because phrasing like “positive at one vertex of S and zero at all others” is the obvious way to express a single-vertex-positive condition, I think it would be perverse or misguided for someone to write “non-negative at all the vertices of S and positive at some vertex” to mean positive at one and only one vertex.
Anyhow, the usual intent of phrasing like “non-negative at all the vertices of the structure S and positive at some vertex” is to describe a function that never is negative, and by dint of going positive somewhere, is non-trivial.
Regarding “some vertices” vs “some vertex”: (1) In mathematical writing one desires to make premises only as strong as necessary for a proof to go through. If we take “some vertices” as implying or suggesting multiple vertices, and “some vertex” as one or more, then the latter condition is weaker, hence desired. (2) In a proof, “some vertex” is likely to be part of a phrase like “some vertex, say
v...” – a construction where “some vertices” would not work.
"of"to make sure the reader does not even by accident misunderstand the phrase
at all. It could be understood in two different ways:
"function is not non-negative"and
"function is non-negative". This is the issue I would like to understand, how to make sure what my pro meant?