A new approach for constructing the graphs determined by their generalized $Q$-spectrum

We define the adjacency matrix $A(G)$ and the degree diagonal matrix $D(G)$ for a graph $G$. The $Q$-spectrum is the multiset of eigenvalues of the $Q$-matrix $Q(G)=A(G)+D(G)$, counted with their algebraic multiplicities. A class of graphs is determined by their generalized $Q$-spectrum ( $\mathrm{D}\mathrm{G}\mathrm{Q}\mathrm{S}$ for short) if any two graphs in the class have the same $Q$-spectrum and so do their complements, implying that they are isomorphic. Tian et al. introduced a new method to construct $\mathrm{D}\mathrm{G}\mathrm{Q}\mathrm{S}$ graphs by considering the rooted product graphs $G\circ P_k$. They proved that when $k=2,3$, $G\circ P_k$ is $\mathrm{D}\mathrm{G}\mathrm{Q}\mathrm{S}$ for a special class of graphs $G$. In this paper, we extend this result and prove that, under the same conditions for $G$, the conclusion holds true for any positive integer $k$. While Wang et al. reported comparable results, our study arrives at these findings through an entirely different methodology. Our proof adopts the elementary properties of the resultants, providing an innovative approach to establishing the $\mathrm{D}\mathrm{G}\mathrm{Q}\mathrm{S}$ property for $G\circ P_k$.