On the dependence polynomial of a graph

The dependence polynomial P G = P G ( z ) of a graph G is defined by P G ( z ) ≔ ∑ i = 0 n ( − 1 ) i c i z i where c i = c i ( G ) is the number of complete subgraphs of G of cardinality i . It is clear that the complete subgraphs of G form a poset relative to subset inclusion. Using Möbius inversion, this yields various identities involving dependence polynomials implying in particular that the dependence polynomial of the line graph L ( G ) of G is determined uniquely by the (multiset of) vertex degrees of G and the number of triangles in G . Furthermore, the dependence polynomial of the complement of the line graph of G is closely related to the matching polynomial of G , one of the most ‘prominent’ polynomials studied in graph theory.