Combinatorial invariants for graph isomorphism problem

TL;DR

This paper proposes a polynomial-time method to compute combinatorial invariants for graph vertices, aiming to fully determine graph isomorphism and potentially impact the P vs NP problem.

Contribution

It introduces a novel approach using invariants like closed path counts to distinguish graphs, with the potential to solve graph isomorphism efficiently.

Findings

Invariants are computed for each vertex based on closed paths.

Sorted invariants can identify isomorphic graphs.

Potential implications for P=NP problem.

Abstract

Presented approach in polynomial time calculates large number of invariants for each vertex, which won't change with graph isomorphism and should fully determine the graph. For example numbers of closed paths of length k for given starting vertex, what can be though as the diagonal terms of k-th power of the adjacency matrix. For k=2 we would get degree of verities invariant, higher describes local topology deeper. Now if two graphs are isomorphic, they have the same set of such vectors of invariants - we can sort theses vectors lexicographically and compare them. If they agree, permutations from sorting allow to reconstruct the isomorphism. I'm presenting arguments that these invariants should fully determine the graph, but unfortunately I can't prove it in this moment. This approach can give hope, that maybe P=NP - instead of checking all instances, we should make arithmetics on these large numbers.