Problem: Suppose a graph $G$ has degree sequence $25, 18, 18, 5, 2, 2$.
How many edges are there in $G$?
I am assuming since the definition of degrees of a vertex that each one of the 6 vertex's (since there are $6$ nums in the degree sequence) has $n$ edges(paths) ($n$ being one of the six numbers in the degree sequence). So would the total number of edges in this case be $25+18+18+5+2+2$?
I don't think this graph exists as a simple graph, but would have to be drawn with the three vertices having degrees of $18$, $18$, and $25$ have loops or multiple edges between them. Do those count in the total number of edges?
Best Answer
Hint: For a vertex $v$, the degree of $v$ is the number of edges incident to $v$. Counting the total degree, you will actually count each edge $\{u,v\}$ twice -- once when you count the degree of $u$, and once when you count the degree of $v$.
So, how should the total degree and the total number of edges be related?
If each edge $\{u,v\}$ gets counted for both $u$ and for $v$, then it has been counted twice -- and so the total degree is just twice the number of edges!
(This works if we allow multiple edges... it works with loops as well, with the convention that a loop on vertex $v$ contributes two to its total degree.)
So, if you notice that your total degree is 70, it must be the case that there are 35 edges.