On the dependence polynomial of a graph

AbstractThe dependence polynomial PG=PG(z) of a graph G is defined by PG(z)≔∑i=0n(−1)icizi where ci=ci(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