A remark on the generalized spectral characterization of the disjoint union of graphs

A graph G is said to be determined by its generalized spectrum (DGS for short) if whenever Γ is a graph such that Γ and G are cospectral with cospectral complements, then Γ is isomorphic to G. Let G∪H be the disjoint union of graphs G and H. In this paper, we give a simple sufficient condition, under which we show that G∪H is DGS if and only if both G and H are DGS. In particular, let H={x} be a singleton graph, we show that if gcd⁡(an,det⁡(W(G)))=1 and an is square-free, then G∪{x} is DGS if and only if G is DGS, where an is the constant term of the characteristic polynomial of G and W(G) is the walk-matrix of G. It is noticed that in Wang and Xu [9], the authors gave a sufficient condition for G∪{x} to be DGS if G is DGS. However, they missed the condition that an is square-free in their theorem, and the result obtained is incorrect. We found a counterexample to their result without this condition and give a correct version of the result accordingly in this paper.