A 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 of a link  
via linking numbers of its components.

  If the link is algebraically split (i.e. all linking 
numbers are zero), the  lowest term can be expressed via 
Milnor's triple linking numbers. Comparing these answers 
we obtain a new matrix-tree theorem, relating enumeration 
of spanning trees in a 3-graph and the Pfaffian of a 
certain skew-symmetric matrix related to it.
  
  This result may be generalized to higher order terms using 
the theory of finite type (Vassiliev) invariants and 
Feynman diagrams.