Determine the number of loops for a given multigraph $G$ ($|V| \geq1$) from Tutte's polynomial $T_G(x,y)$.
Okay, so I tried calculating the total number of edges, which I can get from $T_G(2,2)$, but I'm not sure how to decide which of those are loops. And because the definition of the polynomial doesn't include anything else besides the number of connected components - which I don't see how could help me, I'm not sure how to get further.
I am stuck with this problem, any hint is appreciated.
You can look at the largest power of $y$ which divides $T_G(x,y)$. If you look at the recurrence for the Tutte polynomial you see you multiply by $y$ whenever you have a loop.
Similarly you multiply by $x$ whenever you have a bridge. So, there is a similar (dual) statement for bridges.