Distinctive power of the alliance polynomial for regular graphs

The alliance polynomial of a graph G with order n and maximum degree Δ is the polynomial , 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.