  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
$\mG_M$ where the labels lie in $H^2(BT)$. They also define the subring
$H_T^*(\mG_M)$ of $\bigoplus_{v\in V(\mG_M)}H^*(BT)$, where $V(\mG_M)$ is the
set of vertices of $\mG_M$ and we call $H_T^*(\mG_M)$ the "graph cohomology"
ring of $\mG_M$. 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 $\mG_M$. This gives a new computation of the equivariant cohomology ring
of a flag manifold of type $G_2$.
