标题： 格运算符与图矩阵的等价性 显示英文标题

标题： Equivalence of lattice operators and graph matrices

摘要： 我们探讨格点场论与图论之间的关系，特别强调谱图理论范围内狄拉克和标量格点算子及矩阵之间的相互作用。 除了深入研究谱图理论的基本概念，如邻接矩阵和拉普拉斯矩阵，我们引入了一种名为“反对称化邻接矩阵”的新矩阵，专门针对循环有向图（$T^1$格点）和简单有向路径（$B^1$格点）。 图论矩阵与格点算子之间的非平凡关系表明，图拉普拉斯矩阵反映了格点标量算子和格点费米子中的威尔逊项，而反对称化邻接矩阵及其在高维的扩展则等价于原始格点狄拉克算子。 基于这些联系，我们提供了两个关键断言的严格证明：(i) 自由格点标量算子的零模计数与底层图（格点）的零阶贝蒂数一致。(ii) 自由格点费米子算子中狄拉克零模的最大计数等于当$D$维图由循环有向图（$T^1$格点）和简单有向路径（$B^1$格点）的笛卡尔积产生时所有贝蒂数的累积和。 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).