An improved condition for a family of trees being determined by their generalized spectrum

A graph G is said to be determined by its generalized spectrum (DGS for short), if whenever H is a graph such that H and G are cospectral with cospecral complements then H is isomorphic to G. Let G be an n-vertex graph with adjacency matrix A and W(G)=[e,Ae,…,An−1e] be the walk-matrix of G, where e is the all-one vector. A theorem of Wang [15] shows that if 2−⌊n2⌋det⁡W(G) is odd and square-free, then G is DGS. The above condition is equivalent to that the Smith normal form of W(G) is diag(1,…,1,2,…,2,2d), where d is an odd and square-free integer and the number of 1's appeared in the diagonal is precisely ⌈n2⌉. In this paper, we show that for a tree with irreducible characteristic polynomial over Q, the above oddness and square-freeness assumptions on d can actually be removed, which significantly improves upon the above theorem for trees.