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

Authors: Yonghong Zhang; Ligong Wang

Addresses: School of Mathematics and Statistics, Weinan Normal University, Weinan, Shaanxi – 714099, China; School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, Shaanxi – 710129, China; Xi'an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi'an, Shaanxi – 710129, China ' School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, Shaanxi – 710129, China; Xi'an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi'an, Shaanxi – 710129, China

Abstract: 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_α-matrix of G can be defined as A_α(G) = αD(G) + (1 - α)A(G), where α ∈ [0, 1]. In this paper, our focus is on the linear cyclic polyomino chain F_n and the Möbius 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_α-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_α-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.

Keywords: characteristic polynomial; polyomino chain; circulant matrix; symmetric circulant matrix; spectrum.

DOI: 10.1504/IJES.2025.144933

International Journal of Embedded Systems, 2025 Vol.18 No.1, pp.39 - 49

Received: 24 Sep 2024
Accepted: 09 Dec 2024
Published online: 11 Mar 2025 *

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