Alliance polynomial of regular graphs

The alliance polynomial of a graph $G$ with order $n$ and maximum degree $\mathrm{\Delta }$ is the polynomial $A(G; x) = \sum_{k=-\Delta}^{\Delta} A_{k}(G) , x^{n+k}$,where$A_{k}(G)$isthenumberofexactdefensive$k$-alliancesin$G$\mathrm{.}Weobtainsomepropertiesof$A(G; x)$ 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 $\mathrm{\Delta }$-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not $\mathrm{\Delta }$-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 $10$.