and $A^\dag(G;x)$ are Laurent polynomials, and that the polynomial $A^\dag(G;x)$ does not satisfy any easy variant of Corollary \[c:CubUnim\]. Hence it is convenient to keep the previous choice, although each result in this paper has an analogous for $A^\star(G;x)$.

The following procedure allows to compute the alliance polynomial of a graph $G$ with order $n$, see [@CRST Algorithm 2.1]. This will be used in Section \[sect3\] in order to compute alliance polynomials. Let us consider $W=\{S_1,\dots,S_{2^n-1}\}$ the collection of nonempty subsets of $V$.

\[algorithm\] ${}$Input: adjacency matrix of $G$. Output: alliance polynomial of $G$. The algorithm starts with $A(G;x)=0$ and continues with the following steps, for $1 \le j \le 2^n-1$.

1.  [If $\langle S_j\rangle$ is a connected subgraph, then go to step (2), else replace $j$ by $j+1$ and apply this step again.]{}

2.  [Compute $k_{S_j}$, and add one term $x^{n+k_{S_j}}$ to $A(G;x)$.]{}

3.  [If $j < 2^n-1$, then replace $j$ by $j+1$ and apply step (1) again. If $j = 2^n-1$, then the algorithm stops.]{}

The main aim of this work is to obtain further results about the alliance polynomial of regular graphs (graphs with all vertices with the same degree), since they are a very interesting class 