  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.
