not exist. For instance, for $k\ge 2$ there are no defensive $k$-alliances in the star graph $S_n$ with $n$ vertices. Besides, if $G$ is connected then $V(G)$ is a defensive ${\delta}$-alliance. Notice that for any $S$ with $\langle S\rangle$ connected there exists some $k\in\mathcal{K}$ such that it is a defensive $k$-alliance in $G$.

We define, for $S\subseteq V(G)$ inducing a connected subgraph $\langle S\rangle$, $$\label{eq:k_exact}
    k_{S}:=\displaystyle\max_{} \{k\in\mathcal{K} \,:\,  S \text{ is a defensive $k$-alliance}\}.$$ We say that $k_{S}$ is the *exact index of alliance of* $S$, or also, $S$ is an *exact defensive $k_{S}$-alliance* in $G$. The exact index of alliance of $S$ in $G$ is also $k_{S}= \displaystyle\min_{v\in S} \{{\delta}_{S}(v) - {\delta}_{\overline{S}}(v)\}$.

The *alliance polynomial* of a graph $G$ is defined as an ordinary generating function for the number of exact defensive $k$-alliances: $$\label{eq:Poly1}
    A(G;x)= \displaystyle\sum_{k\in \mathcal{K}} A_{k}(G) x^{n+k}, \ \text{ with } A_{k}(G) \text{ the number of exact defensive } k \text{-alliances in } G.$$

There are several d.p.-equivalent definitions for an alliance polynomial. For instance, another natural definition for alliance polynomials of $G$ could be $A^\star(G;x)= \sum_{k\in \mathcal{K}} A_{k}(G) x^{k}$ (*i.e.*, $A(G;x)= x^n\, A^\star(G;x)$). We may also define it by $A^\dag(G;x)=\sum_{k\in \mathcal{K}} \big(A_k(G)+1\big) x^k$. Note that, $A^\star(G;x)$ 