It means "[The] function is non-negative at all of the vertices in the structure S and positive at some vertex". The of isn't necessary though it may help to clarify the meaning.
There may be some confusion between the phrases not negative at all and non-negative at all. Some examples may help to clarify what I mean. Note that these sentences do not have the same meaning as your original sentence.
This sentence asserts that the function is never negative.
The function is not negative at all.
It could also be written as
The function is not at all negative.
The phrase at all serves to emphasize not.
However in your sentence the phrase at all refers to the set of vertices. It could be written as
The function is not negative at all the vertices ...
(Though, as jwpat points out, there are better ways to say this.)
If it were reordered it would not make sense. Thus at all is not serving to emphasize not.
*The function is not at all negative the vertices ...
This questionquestion discusses the difference between at all and not at all.
