Graph equivariant cohomological rigidity for GKM graphs

TL;DR

This paper establishes a precise correspondence between the isomorphism of GKM graphs and their graph equivariant cohomology algebras, providing a cohomological criterion for graph isomorphism.

Contribution

It introduces a formal definition of GKM graph isomorphism and proves that cohomology algebra isomorphism characterizes graph isomorphism, linking algebraic and combinatorial structures.

Findings

GKM graph isomorphism is characterized by cohomology algebra isomorphism

Provides a cohomological criterion for GKM graph equivalence

Establishes a new connection between graph theory and algebraic topology

Abstract

We formulate the notion of an isomorphism of GKM graphs. We then show that two GKM graphs have isomorphic graph equivariant cohomology algebras if and only if the graphs are isomorphic.