What would a Tutte Polynomial =0 represent?

Question

So I'm working on proving (via contradiction) that the flow number $\phi(G)$ of a bridgeless graph $G$ is always defined. I'm using the flow polynomial, and I got to a point where I have $0=T(0,1-u)$.

So, my question: If $T(x,y)=0$ where $T(x,y)$ is the Tutte polynomial, what does this mean about the graph? Does it mean it has no edges at all?

Thanks!

Answer 1

There are no graphs for which the Tutte polynomial is $0$. One thing that would go wrong if there were such a graph:

• The chromatic polynomial is contained within the Tutte polynomial; if the Tutte polynomial were $0$, then the graph would not be $k$-colourable for all $k \geq 0$. But this is impossible since e.g. we can colour each vertex a different colour.

Answer 2

Actually if you try to color proper a graph with a loop you will always get a zero. So I can't clearly see that 'contained'.

But it is still true, that there is no graph with the zero Tutte polynomial. For example it might be evident from the definition.