Must the score of a graph be positive integer? For graph or multigraph it's obvious, but how about the score of a weighted graph, dose it also have to be positive integers?
(Score of a graph means a degree sequence of the graph G, which is written in non-decreasing order, with the smallest degree come first) 
for example in this graph the score of it is (3,3,3,4,4,5,5,6,6,7),

in this multigraph score of it is (3,4,5).
(and weighted graph here conserns edge-weighted graph, whose edges have been asigned weights, it seems that the degree of a vertex can be non-integer)

in this graph the score is (2,2,3).
reference:
http://mathworld.wolfram.com.
https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms#W
https://www.coursera.org/learn/discrete-mathematics/home/welcome
 A: I've never seen anyone require a weighted graph to have the weights add up to an integer at every vertex. That's not to say it's never happened - but I think it's fair to ask people who do make that assumption to state it explicitly every time they use weighted graphs, and leave the rest of us free not to assume it.
Relatedly, you may have noticed that I feel a bit awkward calling the sum of weights at a vertex "degree" - and only noticed this while rereading my answer. It might be better to refer to the "weighted degree" of a vertex instead, removing all ambiguity. It's possible that some people will use "degree" and "score sequence" to refer to the properties of the underlying unweighted graph, and use "weighted degree" and "weighted score sequence" for the values when weights are taken into account.
Often it's possible to assume without loss of generality that all weighted degrees are integers and in fact that all the weights are integers. For example, if you're considering a random walk on a weighted graph, scaling up all the weights by a factor of $k$ doesn't change the limiting distribution, and scaling them up by a factor of $k$ and then rounding only changes it slightly. (As $k \to \infty$, the change approaches $0$.) 
Of course, if all your weights are integers, you effectively have a multigraph instead, so there's that.
