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Alliance polynomial of regular graphs

The alliance polynomial of a graph $G$ with order $n$ and maximum degree $Δ$ is the polynomial $A(G; x) = \sum_{k=-Δ}^Δ A_{k}(G) \, x^{n+k}$, where $A_{k}(G)$ is the number of exact defensive $k$-alliances in $G$. We obtain some properties of $A(G; x)$ and its coefficients for regular graphs. In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of $Δ$-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not $Δ$-regular. By using this last result and direct computation we find that the alliance polynomial determines uniquely each cubic graph of order less than or equal to $10$.