We construct infinite families of graphs that are determined by their generalized spectrum. This construction is based on new formulae for the determinant of the walk matrix of a graph. The graphs constructed here all satisfy a lower divisibility for the determinant of their walk matrix. 1 spectrum, or a DS graph for short. In [4,8,9] it is conjectured that almost all graphs are DS, more recent surveys can be found in [5].A variant of this problem concerns the generalized spectrum of G which is given by the pair spec(G), spec(G) . In this situation G is said to be determined by its generalized spectrum, or a DGS graph for short, if spec(H), spec(H) = spec(G), spec(G) implies that H is isomorphic to G. For the most recent results on DGS graphs see [13] and [20] in particular. To date only a few families of graphs are known to be DS or DGS. This includes some almost complete graphs [1], pineapple graphs [10] and kite graphs [11]. These are all known to be DS, and in particular DGS. In addition all rose graphs are determined by their Laplacian spectrum except for two specific examples, see [12].In this paper we construct infinite sequences of DGS graphs from certain small starter graphs. This construction involves the walk matrix of a graph and recent results in [20]. Let n be the number of vertices of G and let e be the column vector of height n with all entries equal to 1. Then the walk matrix of G is the n × n matrix W = e, Ae, A 2 e, · · · , A n−1 e