The \(A_{\alpha }\)-Spectrum and \(A_{\alpha }\)-Energy of the Dice Lattice

The dice lattice is a two-dimensional structure derived from hexagonal and triangular lattices, distinguished by its high degree of symmetry and distinctive physical properties. It holds significant relevance in the fields of mathematics, physics, and materials science, particularly in the investigation of topological phenomena and the dynamic behavior of low-dimensional systems. For a given graph G, let A(G), D(G), and Q(G) represent the adjacency matrix, degree matrix, and signless Laplacian matrix of G, respectively. We define

$$\begin{aligned}A_{\alpha }(G) = \alpha D(G) + (1 - \alpha )A(G), \text{ for } \text{ any } \text{ real } \text{ value } \alpha \in [0, 1].\end{aligned}$$

In this paper, we determine the \(A_{\alpha }\)-spectrum and \(A_{\alpha }\)-energy of the dice lattice under toroidal boundary conditions. Furthermore, we utilize these findings to derive the A-spectrum, Q-spectrum, A-energy, and Q-energy of the dice lattice with the same boundary conditions.