The Ring of Graph Invariants - Upper and Lower Bounds for Minimal Generators
نویسنده
چکیده
In this paper we study the ring of graph invariants, focusing mainly on the invariants of simple graphs. We show that all other invariants, such as sorted eigenvalues, degree sequences and canonical permutations, belong to this ring. 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 invariants are required, which are called separator invariants. The minimal set of separator invariants is also the minimal generator set for the ring of graph invariants. We find lower and upper bounds for the minimal number of generator/separator invariants needed for proving graph isomorphism. The minimal number of generators/separators is the transcendence degree of the ring of graph invariants. Finally we find a sufficient condition for Ulam’s conjecture to be true based on Redfield’s enumeration formula.
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