Algebraic invariants of graphs; a study based on computer exploration

TL;DR

This paper explores algebraic invariants of graphs using computer-aided methods, describing the structure of the invariant ring for small graphs and addressing conjectures related to graph isomorphism and reconstruction.

Contribution

The authors analyze the invariant ring of weighted graphs, develop computational tools, and provide new results and counterexamples for small graphs, advancing algebraic approaches to graph isomorphism.

Findings

Complete description of the invariant ring for n<=4

Construction of a generating set for I_5

Counterexamples to conjectures on graph reconstruction

Abstract

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.