The cohomology ring of the GKM graph of a flag manifold of classical type

If a closed smooth manifold $M$ with an action of a torus $T$ satisfies certain conditions, then a labeled graph ${\text{mG}}_M$ with labeling in $H^2(BT)$ is associated with $M$, which encodes a lot of geometrical information on $M$. For instance, the “graph cohomology” ring ${\text{mHT}}^{\ast }({\text{mG}}_M)$ of ${\text{mG}}_M$ is defined to be a subring of ${⨁}_{v\in V({\text{mG}}_M)}{H}^{\ast }(BT)$, where $V({\text{mG}}_M)$ is the set of vertices of ${\text{mG}}_M$, and is known to be often isomorphic to the equivariant cohomology $H_T^{\ast }(M)$ of $M$. In this paper, we determine the ring structure of ${\text{mHT}}^{\ast }({\text{mG}}_M)$ with $\mathbb{Z}$ (resp. $\mathbb{Z}[1\mathrm{/}2]$) coefficients when $M$ is a flag manifold of type A, B or D (resp. C) in an elementary way.