The graph cohomology ring of the GKM graph of a flag manifold of type $G_2$

Suppose a compact torus $T$ acts on a closed smooth manifold $M$. Under certain conditions, Guillemin and Zara associate to $(M,T)$ a labeled graph ${\text{mG}}_M$ where the labels lie in $H^2(BT)$. They also define the subring $H_T^{\ast }({\text{mG}}_M)$ 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 we call $H_T^{\ast }({\text{mG}}_M)$ the “graph cohomology” ring of ${\text{mG}}_M$. It is known that the equivariant cohomology ring of $M$ can be described by using combinatorial data of the labeled graph. The main result of this paper is to determine the ring structure of equivariant cohomology ring of a flag manifold of type $G_2$ directly, using combinatorial techniques on the graph ${\text{mG}}_M$. This gives a new computation of the equivariant cohomology ring of a flag manifold of type $G_2$.