The Induced Subgraph Order on Unlabelled Graphs

A differential poset is a partially ordered set with raising and lowering operators U and D which satisfy the commutation relation DU−UD = rI for some constant r. This notion may be generalized to deal with the case in which there exist sequences of constants {qn}n≥0 and {rn}n≥0 such that for any poset element x of rank n, DU(x) = qnUD(x)+rnx. Here, we introduce natural raising and lowering operators such that the set of unlabelled graphs, ordered by G ≤ H if and only if G is isomorphic to an induced subgraph of H, is a generalized differential poset with qn = 2 and rn = 2 . This allows one to apply a number of enumerative results regarding walk enumeration to the poset of induced subgraphs.