Tutte Polynomial

Let $G$ be a Graph, and let $\mathop{\rm ea}(T)$ denote the cardinality of the set of externally active edges of a spanning tree $T$ of $G$ and $\mathop{\rm ia}(T)$ denote the cardinality of the set of internally active edges of $T$. Then

$$t_G(x,y)=\sum_{T\subseteq G} x^{\mathop{\rm ia}(T)}y^{\mathop{\rm ea}(T)}.$$

References

Gessel, I. M. and Sagan, B. E. ``The Tutte Polynomial of a Graph, Depth-First Search, and Simplicial Complex Partitions.'' Electronic J. Combinatorics 3, No. 2, R9, 1-36, 1996. http://www.combinatorics.org/Volume_3/volume3_2.html#R9.

Tutte, W. T. ``A Contribution to the Theory of Chromatic Polynomials.'' Canad. J. Math. 6, 80-91, 1953.