Algebraic invariants of graphs; a study based on computer exploration

We consider the ring I_n of polynomial invariants over weighted graphs on n vertices. Our primary interest is the use of this ring to define and explore algebraic versions of isomorphism problems of graphs, such as Ulam's reconstruction conjecture. There is a huge body of literature on invariant theory which provides both general results and algorithms. However, there is a combinatorial explosion in the computations involved and, to our knowledge, the ring I_n has only been completely described for n<=4. This led us to study the ring I_n in its own right. We used intensive computer exploration for small n, and developed PerMuVAR, a library for MuPAD, for computing in invariant rings of permutation groups. We present general properties of the ring I_n, as well as results obtained by computer exploration for small n, including the construction of a medium sized generating set for I_5. We address several conjectures suggested by those results (low degree system of parameters, unimodality), for I_n as well as for more general invariant rings. We also show that some particular sets are not generating, disproving a conjecture of Pouzet related to reconstruction, as well as a lemma of Grigoriev on the invariant ring over digraphs. We finally provide a very simple minimal generating set of the field of invariants.