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 = − 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 . [ABSTRACT FROM AUTHOR]