The Algebra of Graph Invariants - Lower and Upper Bounds for Minimal Generators

TL;DR

This paper explores the algebraic structure of graph invariants, establishing bounds on the minimal number of invariants needed for graph isomorphism and proposing a condition related to Ulam's conjecture.

Contribution

It provides new lower and upper bounds for the minimal generator set of graph invariants and links these findings to Ulam's conjecture.

Findings

Established bounds for minimal generator invariants

Identified a sufficient condition for Ulam's conjecture

Connected algebraic invariants to graph isomorphism proofs

Abstract

In this paper we study the algebra of graph invariants, focusing mainly on the invariants of simple graphs. All other invariants, such as sorted eigenvalues, degree sequences and canonical permutations, belong to this algebra. In fact, every graph invariant is a linear combination of the basic graph invariants which we study in this paper. To prove that two graphs are isomorphic, a number of basic invariants are required, which are called separator invariants. The minimal set of separator invariants is also the minimal basic generator set for the algebra of graph invariants. We find lower and upper bounds for the minimal number of generator/separator invariants needed for proving graph isomorphism. Finally we find a sufficient condition for Ulam's conjecture to be true based on Redfield's enumeration formula.