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}n0 and {rn}n0 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 = 2n. This allows one to apply a number of enumerative results regarding walk enumeration to the poset of induced subgraphs.

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: Craig A. Sloss
• Keywords: COMBINATORICS 2006, Commutation Relation, Operators, Poset Element