  Let $\Gamma=(V,E)$ be a simple graph. For a nonempty set $X\subseteq V$, and
a vertex $v\in V$, $\delta_{X}(v)$ denotes the number of neighbors $v$ has in
$X$. A nonempty set $S\subseteq V$ is a \emph{defensive $k$-alliance} in
$\Gamma=(V,E)$ if $\delta_S(v)\ge \delta_{\bar{S}}(v)+k,$ $\forall v\in S.$ A
defensive $k$-alliance $S$ is called \emph{global} if it forms a dominating
set. The \emph{global defensive $k$-alliance number} of $\Gamma$, denoted by
$\gamma_{k}^{a}(\Gamma)$, is the minimum cardinality of a defensive
$k$-alliance in $\Gamma$. We study the mathematical properties of
$\gamma_{k}^{a}(\Gamma)$.
