Distance (signless) Laplacian spectra and energies of two classes of cyclic polyomino chains

Let D(G) and Tr(G) be the distance matrix and the diagonal matrix of vertex transmissions of a graph G, respectively. The distance Laplacian matrix and the distance signless Laplacian matrix of G are defined as D^L(G)=Tr(G)−D(G) and D^Q(G)=Tr(G)+D(G), respectively. In this paper, we consider the distance Laplacian spectra and the distance signless Laplacian spectra of the linear cyclic polyomino chain F_n and the Möbius cyclic polyomino chain M_n. By utilizing the properties of circulant matrices, we give the characteristic polynomials and the eigenvalues for the distance Laplacian matrices and the distance signless Laplacian matrices of the graphs F_n and M_n, respectively. Furthermore, we provide the exactly values of the distance Laplacian energy and the distance signless Laplacian energy of the graph F_n, and the upper bounds on the distance Laplacian energy and the distance signless Laplacian energy of the graph M_n, respectively.