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 <Equation ID="Equ24"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3523_Article_Equ24.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="457" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned}A_{\alpha }(G) = \alpha D(G) + (1 - \alpha )A(G), \text{ for } \text{ any } \text{ real } \text{ value } \alpha \in [0, 1].\end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>A</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi mathvariant="normal">for</mi> <mi mathvariant="normal">any</mi> <mi mathvariant="normal">real</mi> <mi mathvariant="normal">value</mi> <mi>α</mi> <mo>∈</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper, we determine the <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3523_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$A_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-spectrum and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10955_2025_3523_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$A_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>-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.

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

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发表日期：

2025年10月4日

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Xiaxia Zhang,
Xiaoling Ma

摘要

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 $\begin{aligned}A_{\alpha }(G) = \alpha D(G) + (1 - \alpha )A(G), \text{ for } \text{ any } \text{ real } \text{ value } \alpha \in [0, 1].\end{aligned}$ $\begin{matrix} A_{α} (G) = α D (G) + (1 - α) A (G), for any real value α \in [0, 1] . \end{matrix}$ In this paper, we determine the $A_{\alpha }$ $A_{α}$ -spectrum and $A_{\alpha }$ $A_{α}$ -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.

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

摘要