$A_{\alpha}$-Spectra of $Q$- and $T$-Join Graphs with Applications to Cospectral Constructions

For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of a graph $G$ is defined by $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal degree matrix of $G$, respectively. In this paper, we study the $A_{\alpha}$-characteristic polynomials and $A_{\alpha}$-spectra of graphs obtained via four recently introduced join operations, namely the $Q$-vertex join, $Q$-edge join, $T$-vertex join, and $T$-edge join, applied to two graphs $G_1$ and $G_2$. We derive explicit expressions for the $A_{\alpha}$-characteristic polynomials of these constructions when the first factor graph is regular. Furthermore, we determine the complete $A_{\alpha}$-spectra of these graphs in terms of the $A_{\alpha}$-spectra of the factor graphs, particularly when the second factor graph is regular or complete bipartite. The significance of these results lies in the fact that they enable efficient computation of the $A_{\alpha}$-spectra of large complex graphs arising from these joins, directly from the $A_{\alpha}$-spectra of the smaller constituent graphs, without explicitly constructing and handling the complex $A_{\alpha}$-matrices of those large graphs. Finally, as an application, we demonstrate how to construct infinitely many families of non-isomorphic graphs that are $A_{\alpha}$-cospectral.