  Let $\Gamma$ be a finite d-valent graph and G an n-dimensional torus. An
``action'' of G on $\Gamma$ is defined by a map, $\alpha$, which assigns to
each oriented edge e of $\Gamma$ a one-dimensional representation of G (or,
alternatively, a weight, $\alpha_e$, in the weight lattice of G). For the
assignment, $e \to \alpha_e$, to be a schematic description of a ``G-action'',
these weights have to satisfy certain compatibility conditions: the GKM axioms.
We attach to $(\Gamma, \alpha)$ an equivariant cohomology ring,
$H_G(\Gamma)=H(\Gamma,\alpha)$. By definition this ring contains the
equivariant cohomology ring of a point, $\SS(\fg^*) = H_G(pt)$, as a subring,
and in this paper we will use graphical versions of standard Morse theoretical
techniques to analyze the structure of $H_G(\Gamma)$ as an $\SS(\fg^*)$-module.
