# Structure of Graph Posets for Orders 4 to 8

MacDougall, James A. and Eggleton, Roger B. and Adams, Peter D. (2004) Structure of Graph Posets for Orders 4 to 8. Congressus Numerantium, 166 . pp. 63-81.

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The poset $G(n)$ comprises the unlabelled simple graphs of order $n$, with partial ordering $G \leq H$ whenever $G$ is a spanning subgraph of $H$. We define a modified Steinbach numbering of the graphs in $G(n)$, apply this numbering to each $G(n)$ with $n \leq 8$, and use it to tabulate the Hasse diagram structure of the posets with $4 \leq n \leq 8$ together with key aspects of the independence structure of these posets. In particular, the Hasse diagram of $G(8)$ is a directed graph of order 12346 and size 125066; the poset $G(8)$ has 51952895 independent pairs of graphs, and 96775426396 independent triples. We present 14 tables of descriptive data for $G(n)$ with $4 \leq n \leq 8$. All of the underlying data can be found on our webpage www.maths.uq.edu.au/~pa/research/posets4to8.html