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 A(alpha)-matrix of G can be defined as A(alpha)(G) = alpha D(G) + (1 - alpha)A(G), where alpha is an element of [0, 1]. In this paper, our focus is on the linear cyclic polyomino chain F-n and the M & ouml;bius cyclic polyomino chain M-n. 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 A(alpha)-matrix of the graphs F-n and M-n, respectively. Furthermore, we provide the exact values of the Laplacian energies and the signless Laplacian energies of two graphs F-n and M-n, respectively. Finally, the upper bounds on the A(alpha)-energies of the graphs F-n and M-n 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 F-n and M-n may provide a new perspective for designing quantum computing models or understanding the complex interactions in quantum systems.