We introduce a new bivariate polynomial which we call the defensive alliance
polynomial and denote it by da(G; x, y). It is a generalization of the alliance
polynomial [Carballosa et al., 2014] and the strong alliance polynomial
[Carballosa et al., 2016]. We show the relation between da(G; x, y) and the
alliance, the strong alliance and the induced connected subgraph [Tittmann et
al., 2011] polynomials. Then, we investigate information encoded in da(G; x, y)
about G. We discuss the defensive alliance polynomial for the path graphs, the
cycle graphs, the star graphs, the double star graphs, the complete graphs, the
complete bipartite graphs, the regular graphs, the wheel graphs, the open wheel
graphs, the friendship graphs, the triangular book graphs and the quadrilateral
book graphs. Also, we prove that the above classes of graphs are characterized
by its defensive alliance polynomial. A relation between induced subgraphs with
order three and both subgraphs with order three and size three and two
respectively, is proved to characterize the complete bipartite graphs. Finally,
we present the defensive alliance polynomial of the graph formed by attaching a
vertex to a complete graph. We show two pairs of graphs which are not
characterized by the alliance polynomial but characterized by the defensive
alliance polynomial
