Dα-characteristic polynomials and Dα-energies of two classes of cyclic polyomino chains

For a connected graph [Formula: see text], let [Formula: see text] and [Formula: see text] denote the distance matrix and the diagonal matrix of vertex transmissions of [Formula: see text], respectively. The generalized distance matrix [Formula: see text] of [Formula: see text] is defined as [Formula: see text], where [Formula: see text]. In this paper, we investigate the [Formula: see text]-spectra of the linear cyclic polyomino chain [Formula: see text] and the Möbius cyclic polyomino chain [Formula: see text]. By using the properties of circulant matrices, the characteristic polynomials and the eigenvalues for the distance matrices and the [Formula: see text]-matrices of the graphs [Formula: see text] and [Formula: see text] are given, respectively. Furthermore, the precise values on the distance energy and the [Formula: see text]-energy of the graph [Formula: see text] are presented. Additionally, the upper bounds on the distance energy and the [Formula: see text]-energy of the graph [Formula: see text] are established.