Let $G$ be a multigraph, i.e, there can be more than one edge between a pair of vertices. It is clear that the chromatic polynomial cannot capture these multi-edges. Because chromatic polynomial just cares whether two vertices are adjacent or not and doesn't care about the number of edges between them.

My question: is there any combinatorial tool like chromatic polynomial which captures the number of edges between the vertices as well?

Kindly share your thoughts.

Thank you.