Is a graph simple, given the number of vertices and the degree sequence? Does there exist a simple graph with five vertices of the following degrees?
(a) 3,3,3,3,2
I know that the answer is no, however I do not know how to explain this.
(b) 1,2,3,4,3
No, as the sum of the degrees of an undirected graph is even.
(c) 1,2,3,4,4
Again, I believe the answer is no however I don't know the rule to explain why.
(d) 2,2,2,1,1
Same as above.
What method should I use to work out whether a graph is simple, given the number of vertices and the degree sequence? 

(a) 3,3,3,3,2   - YES!  Graph Justifies claim
(b)1,2,3,4,3   - NO -Follows from the Handshaking Lemma
(c)1,2,3,4,4   - ANYBODY? (has no problem by Handshaking Lemma)
(d)2,2,2,1,1  - YES! Graph Justifies Claim
