  The aim of this paper is to give an upper bound for the dimension of a torus
$T$ which acts on a GKM manifold $M$ effectively. In order to do that, we
introduce a free abelian group of finite rank, denoted by
$\mathcal{A}(\Gamma,\alpha,\nabla)$, from an (abstract) $(m,n)$-type GKM graph
$(\Gamma,\alpha,\nabla)$. Here, an $(m,n)$-type GKM graph is the GKM graph
induced from a $2m$-dimensional GKM manifold $M^{2m}$ with an effective
$n$-dimensional torus $T^{n}$-action, say $(M^{2m},T^{n})$. Then it is shown
that $\mathcal{A}(\Gamma,\alpha,\nabla)$ has rank $\ell(> n)$ if and only if
there exists an $(m,\ell)$-type GKM graph $(\Gamma,\widetilde{\alpha},\nabla)$
which is an extension of $(\Gamma,\alpha,\nabla)$. Using this necessarily and
sufficient condition, we prove that the rank of
$\mathcal{A}(\Gamma,\alpha,\nabla)$ for the GKM graph of $(M^{2m},T^{n})$ gives
an upper bound for the dimension of a torus which can act on $M^{2m}$
effectively. As an application, we compute the rank of
$\mathcal{A}(\Gamma,\alpha,\nabla)$ of the complex Grassmannian of $2$-planes
$G_{2}(\mathbb{C}^{n+2})$ with some effective $T^{n+1}$-action, and prove that
the $T^{n+1}$-action on $G_{2}(\mathbb{C}^{n+2})$ is the maximal effective
torus action.
