The classical theorem of Kirchhoff expresses the number of spanning trees of a graph via the determinant of some matrix related to its adjacency matrix. A knot-theoretical version of this theorem relates two formulas expressing the lowest term of the Alexander-Conway polynomial $C(L)$ of a link $L$ via linking numbers of its components.

If $L$ is algebraically split (i.e. all linking numbers are zeros), the lowest term of the $C(L)$ can be expressed via Milnor higher linking numbers in two different ways. Comparing these answers we obtain a new matrix-tree theorem, relating enumeration of spanning trees in a 2-graph and the Pfaffian of a certain skew-symmetric matrix related to it.