The Induced Subgraph Order on Unlabelled Graphs

Abstract

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 $\{q_n\}_{n\geq 0}$ and $\{r_n\}_{n\geq 0}$ such that for any poset element $x$ of rank $n$, $DU(x) = q_n UD(x) + r_nx$. Here, we introduce natural raising and lowering operators such that the set of unlabelled graphs, ordered by $G\leq H$ if and only if $G$ is isomorphic to an induced subgraph of $H$, is a generalized differential poset with $q_n=2$ and $r_n = 2^n$. This allows one to apply a number of enumerative results regarding walk enumeration to the poset of induced subgraphs.