标题： 格算子与图矩阵的等价性 显示英文标题

标题： Equivalence of lattice operators and graph matrices

摘要： 我们探讨了格点场论与图论之间的关系，并特别强调了狄拉克算符和标量格点算符与谱图论中矩阵之间的相互作用。除了深入研究谱图论的基本概念（如邻接矩阵和拉普拉斯矩阵），我们还引入了一种新的矩阵，称为“反对称化邻接矩阵”，专门针对循环有向图（$T^1$格点）和简单有向路径（$B^1$格点）。 图论矩阵与格点算符之间非平凡的关系表明，图拉普拉斯矩阵反映了格点标量算符以及格点费米子中的威尔逊项，而反对称化邻接矩阵及其高维扩展则等价于简单的格点狄拉克算符。 基于这些联系，我们对两个关键断言提供了严格的证明：(i) 自由格点标量算符的零模数量与底层图（格点）的零阶贝蒂数一致。(ii) 自由格点费米子算符中狄拉克零模的最大数量等价于当$D$维图是由循环有向图（$T^1$格点）和简单有向路径（$B^1$格点）的笛卡尔积生成时，所有贝蒂数的总和。 显示英文摘要

摘要： We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix named as "anti-symmetrized adjacency matrix", specifically tailored for cycle digraphs ($T^1$ lattice) and simple directed paths ($B^1$ lattice). The nontrivial relation between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the anti-symmetrized adjacency matrix, along with its extensions to higher dimensions, are equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the $D$-dimensional graph results from a cartesian product of cycle digraphs ($T^1$ lattice) and simple directed paths ($B^1$ lattice).