Charcoal, 104 bytes

⊞υθＦυ«≔§ι⁰η≔⊟ιζ¿ζ«≔⊟ζδ≔ＥζＥκ⎇⁼μ⌈δ⌊δμεＦ⟦⟦ηζ⟧⟦⊖ηΦε⁼λ⌕εκ⟧⟧«⊞υκ⊞ικ»»»Ｆ⮌υ¿⊖Ｌι«≔⊟⊟ιη⊞ιＥ⊟⊟ι⁻κ§ηλ»⊞ιＥ⊕§θ⁰⁼κ§ι⁰Ｉ⊟θ⊞υθＦυ«≔§ι⁰η≔⊟ιζ¿ζ«≔⊟ζδ≔⁻ζ⟦δ⮌δ⟧ζＦ⟦⟦ηζ⟧⟦⊖ηＥζＥκ⎇⁼μ⌈δ⌊δμ⟧⟧«⊞υκ⊞ικ»»»Ｆ⮌υ¿⊖Ｌι«≔⊟⊟ιη⊞ιＥ⊟⊟ι⁻κ§ηλ»⊞ιＥ⊕§θ⁰⁼κ§ι⁰Ｉ⊟θ 

Try it online!Try it online! Link is to verbose version of code. Takes input as a pair of the number of vertices and a list of edges and outputs the polynomial as a list of coefficients in ascending order. Explanation:

⊞υθＦυ« 

Start processing with the input graph.

≔§ι⁰η≔⊟ιζ 

Get the number of vertices and list of edges of this graph.

¿ζ« 

If the graph still has edges, then...

≔⊟ζδ 

Remove an edge from the list.

≔ＥζＥκ⎇⁼μ⌈δ⌊δμε≔⁻ζ⟦δ⮌δ⟧ζ 

Calculate the result of merging the two verticesRemove any other duplicate copies of that edge togetherfrom the list.

Ｆ⟦⟦ηζ⟧⟦⊖ηΦε⁼λ⌕εκ⟧⟧«Ｆ⟦⟦ηζ⟧⟦⊖ηＥζＥκ⎇⁼μ⌈δ⌊δμ⟧⟧« 

Loop over the graph with the edge removed and the graph with the edge merged and deduplicated.

⊞υκ⊞ικ 

Push each graph both to the list of graphs to process but also to the graph that is being processed.

»»»Ｆ⮌υ 

Loop over the list of graphs in reverse order.

¿⊖Ｌι« 

If this graph had to be split into two simpler graphs, then...

≔⊟⊟ιη⊞ιＥ⊟⊟ι⁻κ§ηλ 

... push the difference between the two polynomials from the simpler graphs to this graph, ...

»⊞ιＥ⊕§θ⁰⁼κ§ι⁰ 

... otherwise push the polynomial for a trivial graph to this graph.

Ｉ⊟θ 

Output the polynomial for the original graph.