I am trying to understand if I am thinking about this questions correctly. For each of the following, either give the edge set of a graph on vertex set {a,b,c} that has the stated degree sequence, or show that no such graph exists.
a) (0,0,0) - I said {a,b,c}
b) (1,1,0) - I said {a-b, c}
c) (1,1,1)
d) (2,0,0)
e) (2,2,2)
I am pretty confident on the first two but I'm not so sure on the last three. My intuition is that no such graphs exists for the last three as the degree sequence is not possible with the three nodes listed. I tried to picture the graphs visually.
Learning graph theory for the first time so any help would be great!
(a) (b) Your answers are OK.
(c) Such a degree sequence is impossible, because a sum of degrees of vertices of a graph is twice its number of edges, so it is an even number, which fails for the given sequence.
(d) Such a degree sequence is impossible, because $\deg a>0$ implies $\deg b>0$ or $\deg c>0$, which fails for the given sequence.
(e) Answered in comments.
In general, a necessary and sufficient condition when a sequence $a_1\le a_2\le \dots\le a_n$ of non-negative integers can be represented as the degree sequence of a finite simple graph on $n$ vertices is provided by the Erdős–Gallai theorem: the sum $a_1+a_2+\dots+a_n$ is even and for each $1\le k\le n$ holds $$\sum_{i=1}^k a_i\le k(k-1)+ \sum_{i=k+1}^n \min\{a_i, k\}.$$