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

For a connected graph G, let D(G) and Tr(G) denote the distance matrix and the diagonal matrix of vertex transmissions of G, respectively. The generalized distance matrix D_α(G) of G is defined as D_α(G) = αTr(G) + (1 − α)D(G), where α ∈ [0, 1]. In this paper, we investigate the D_α-spectra of the linear cyclic polyomino chain L_n and the Möbius cyclic polyomino chain M_n. By using the properties of circulant matrices, the characteristic polynomials and the eigenvalues for the distance matrices and the D_α-matrices of the graphs L_n and M_n are given, respectively. Furthermore, the precise values on the distance energy and the D_α-energy of the graph L_n are presented. Additionally, the upper bounds on the distance energy and the D_α-energy of the graph M_n are established.