Spectral Characterizations of the Aα−-Matrix for Graph Products

For α∈0,1, the matrix Aα−G=αDG+α−1AG defines a Laplacian–type operator associated with a graph G, interpolating between −AG and DG. In this paper, we develop a systematic study of the Aα−-spectrum under several classical graph products. Complete spectral characterizations are obtained for the Cartesian, lexicographic, and generalized lexicographic products. For the direct and strong products of connected regular graphs, explicit formulas are derived in terms of the Laplacian and signless Laplacian spectra of the factor graphs. The results demonstrate that Aα− exhibits a spectral behavior that is structurally distinct from the classical Aα-matrix, particularly in the presence of product constructions. These findings provide a unified framework for the analysis of Aα− across graph products and contribute to the broader development of parametrized spectral graph theory.