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If a closed smooth manifold $M$ with an action of a torus $T$ satisfiescertain conditions, then a labeled graph $\mG_M$ with labeling in $H^2(BT)$ isassociated with $M$, which encodes a lot of geometrical information on $M$. Forinstance, the "graph cohomology" ring $\mHT^*(\mG_M)$ of $\mG_M$ is defined tobe a subring of $\bigoplus_{v\in V(\mG_M)}H^*(BT)$, where $V(\mG_M)$ is the setof vertices of $\mG_M$, and is known to be often isomorphic to the equivariantcohomology $H^*_T(M)$ of $M$. In this paper, we determine the ring structure of$\mHT^*(\mG_M)$ with $\Z$ (resp. $\Z[1/2]$) coefficients when $M$ is a flagmanifold of type A, B or D (resp. C) in an elementary way.

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