On the dependence polynomial of a graph

The dependence polynomial PG=PG(z)PG=PG(z) of a graph GG is defined by PG(z)≔∑i=0n(−1)icizi where ci=ci(G)ci=ci(G) is the number of complete subgraphs of GG of cardinality ii. It is clear that the complete subgraphs of GG 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)L(G) of GG is determined uniquely by the (multiset of) vertex degrees of GG and the number of triangles in GG. Furthermore, the dependence polynomial of the complement of the line graph of GG is closely related to the matching polynomial of GG, one of the most ‘prominent’ polynomials studied in graph theory.