On the characteristic polynomials and the spectra of two classes of cyclic polyomino chains

Polyhedral graphs hold significant importance in graph theory as well as in other diverse fields. In graph theory, they serve as fundamental objects for understanding various structural properties and topological characteristics. Let A(G) and D(G) be the adjacency matrix and the diagonal matrix of vertex degrees of a graph G, respectively. The Laplacian matrix of G is denoted as L(G) = D(G) − A(G), while the signless Laplacian matrix of G is denoted as Q(G) = D(G) + A(G). Additionally, the Aα-matrix of G can be defined as Aα(G) = αD(G) + (1 − α)A(G), where α ∈ [0, 1]. In this paper, our focus is on the linear cyclic polyomino chain Fn and the Möbius cyclic polyomino chain Mn. By utilising the computational method of the determinant of a circulant matrix, we present the characteristic polynomials and eigenvalues of the Laplacian matrix, the signless Laplacian matrix, and the Aα-matrix of the graphs Fn and Mn, respectively. Furthermore, we provide the exact values of the Laplacian energies and the signless Laplacian energies of two graphs Fn and Mn, respectively. Finally, the upper bounds on the Aα-energies of the graphs Fn and Mn are given, respectively. In quantum physics, the spectral properties of graphs can be associated with quantum states and energy levels. The research results of the graphs Fn and Mn may provide a new perspective for designing quantum computing models or understanding the complex interactions in quantum systems.