Does this graph exist
" 3 nodes of degree 0 1 3 respectively"
i don't think so because if you have a degree of zero it would be disconnected from the other 2 nodes and on that premise you cannot have 3 degrees with 2 nodes, but could someone please help me confirm or deny this.
No if we only consider simple graphs, yes otherwise.
Simple Graph Case:
If one of the nodes has degree three, that means it is connected to three other nodes by edges. So the graph must have at least four nodes.
But we know it only has three. So it's impossible for the graph to exist (because four is more than three).
Non-Simple Graph Case:
Consider a graph with three nodes $a,b,c$. Let $a$ have degree zero. Let $b$ share an edge with $c$. Add a loop on $b$ (an edge from $b$ to $b$).
Then $a$ has degree $0$, $b$ has degree $3$, and $c$ has degree $1$.
It's a useful fact that a node with a loop on it has degree $2$.
The proposition is also trivially true if we allow multigraphs (graphs with multiple edges between nodes).