The coboundary polynomial is a bivariate graph polynomial which can be expressed in terms of the Tutte polynomial of a graph by

where is the connected component count and is the vertex count of a graph (Martin and Reiner 2005; Ardila 2007).

The coboundary polynomial provides a particularly concise way of expression generating functions for the Tutte polynomial of a complete graph or complete bipartite graph .

More things to try:

Weisstein, Eric W. "Coboundary Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CoboundaryPolynomial.html