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)=∑k=−ΔΔAk(G)xn+k, where Ak(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.