$G$ with minimum degree ${\Delta}$; hence, $A_{{\Delta}}(G)$ is the number of connected components of $G$. Besides, since any connected component has cardinality greater than ${\Delta}$, we obtain the upper bound of $A_{{\Delta}}(G)$.]{}

v)  [If $u,v\in V$ with $u\sim v$, then $S:=\{u,v\}$ is an exact defensive ($2-{\Delta}$)-alliance since $1={\delta}_S(u)={\delta}_{\overline{S}}(u)+2-{\Delta}$ and $1={\delta}_S(v)={\delta}_{\overline{S}}(v)+2-{\Delta}$. Thus, we obtain $A_{-{\Delta}+2}(G)\ge m$. Note that if ${\Delta}=1$, we have the second inequality. Assume that ${\Delta}\ge2$. Without loss of generality we can assume that $G$ is connected; otherwise, it suffices to analyze each connected component of $G$. Let us define $S_v:=V\setminus\{v\}$ for any $v\in V$. Since ${\delta}_{S_v}(u)\ge {\Delta}-1$, ${\delta}_{\overline{S_v}}(u)\le1$ for every $u\in S_v$ and both equalities hold for every $w\in N(v)$, we have that $S_v$ is an exact defensive (${\Delta}-2$)-alliance if $v$ is a non-cut vertex, or contains at least two exact defensive (${\Delta}-2$)-alliances if $v$ is a cut vertex.]{}

vi) [The first statement is a consequence of Theorem \[p:AlliPoly\] iv). Consider an exact defensive $k$-alliance $S$ in $G$. So, there exists $v\in S$ with $$2{\delta}_{S}(v) = {\delta}(v) + k = {\Delta}+k.$$ Then, ${\Delta}\equiv k \ (\text{mod } 2)$, $n + k \equiv n + {\Delta}\ (\text{mod } 2)$ and we have the result.]{}

vii) [Since $Deg_{\min}\big(A(G;x)\big)=n-{\Delta}$, we have 