Explain this theorem by Hakimi? This is a theorem by S L Hakimi.
Can anyone explain what this theorem is trying to say?
Theorem 1. The necessary and sufficient conditions for positive integers $d_1,d_2,\cdots, d_n$ to be realizable ( as the degrees of the vertices of a linear graph ) are : 
(i) $ \sum_{i=1}^nd_i=2e$, where e is an integer,
(ii) $ \sum_{i=1}^{n-1}d_i \geq d_n$.
 A: I'm guessing "linear graph" means "multigraph without loops" and the integers are arranged in non-decreasing order - otherwise this theorem does not seem realistic.
The degree of a vertex is the number of edges in the graph that have an endpoint on that vertex. Given a set of points, we can ask what the possible combinations of degrees (or degree sequences) are for a graph using those points as vertices; and in particular we can assign an integer to each point and ask whether there is a way to draw edges between the points such that the degree of each vertex is the number we assigned to it.
Now, it should be fairly obvious that the two given conditions are necessary for such a graph to exist - given such a graph we can conclude them quite easily:
(i) Every edge contributes $1$ to the degree of each of its two endpoints, and thus the sum of degrees is always twice the edge count.
(ii) Given that we are not allowed loops, the degree $d_n$ of the last vertex must be at most the sum of all the other degrees, since each of the $d_n$ edges attached it contributes $1$ to the degree of some other vertex.
Thus, the main content of the theorem is the converse - the claim that if (i) and (ii) hold then there is a graph with the desired degrees.
