Algebraic invariants of graphs; a study based on computer exploration

We consider the ring Jn of polynomial invariants overweighted graphs on n vertices. Our primary interest is the use ofthis ring to define and explore algebraic versions of isomorphismproblems of graphs, such as Ulam's reconstruction conjecture. There is a huge body of literature on invariant theory whichprovides both general results and algorithms. However, there is acombinatorial explosion in the computations involved and, to ourknowledge, the ring Jn has only been completelydescribed for n ≤ 4. This led us to study the ring Jn in its own right. Weused intensive computer exploration for small n, and developedPerMuVAR, a library for MuPAD, for computing in invariant rings ofpermutation groups. We present general properties of the ring Jn, as wellas results obtained by computer exploration for small n, includingthe construction of a medium sized generating set forJn. We address several conjectures suggested by thoseresults (low degree system of parameters, unimodality), forJn as well as for more general invariant rings. We alsoshow that some particular sets are not generating, disproving aconjecture of Pouzet related to reconstruction, as well as a lemmaof Grigoriev on the invariant ring over digraphs. We finallyprovide a very simple minimal generating set of the field ofinvariants.