I was wondering if you could help verify if my answer is correct. The question is:
Is (1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 4, 5) a degree sequence of a graph?
Is it a degree sequence of a tree?
My answers are:
Yes it is a graph because the sum of the degrees / 2 is 12 and even, therefore making it a graph.
No, it is not a tree because the tree satisfies $|V| = |E| - 1$ and $12 - 1 = 11$. There are 13 vertices in total. Are these right?
UPDATED
(2) is not correct, this potentially could be a tree. For the sequence to be a tree you need $|V| = |E| + 1$, not as you wrote. But you have the sum of all degrees is $S=24$, so $|E| = S/2 = 12$ and $|V| = 13$. Can you construct the actual tree?
For (1) you did not prove it is a graph. Can you construct a graph with such a sequence? Showing one would be a proof...