This is the sequel to the author’s previous paper  with Matthias Franz. In the present paper, we introduce the notion of equivariant total Chern class of a GKM graph and show that the pair of graph equivariant cohomology and the equivariant total Chern class determines the GKM graph completely. We also show that for a torus graph in the sense of Maeda–Masuda–Panov, the pair of graph equivariant cohomology and the equivariant 1-st Chern class determines the torus graph completely.
"On the sign ambiguity in equivariant cohomological rigidity of GKM graphs." Proc. Japan Acad. Ser. A Math. Sci. 97 (9) 76 - 81, November 2021. https://doi.org/10.3792/pjaa.97.015