The GraphBLAS community has demonstrated the power of linear algebra-leveraged graph algorithms, such as matrix-vector products for breadth-first search (BFS) traversals. This paper investigates the algebraic conditions needed for such computations when working with directed hypergraphs, represented by incidence arrays with entries from an arbitrary value set with binary addition and multiplication operations. Our results show the one-step BFS traversal is equivalent to requiring specific algebraic properties of those operations. Assuming identity elements 0, 1 for operations, we show that the two operations must be zero-sum-free, zero-divisor-free, and 0 must be an annihilator under multiplication. Additionally, associativity and commutativity are shown to be necessary and sufficient for independence of the one-step BFS computation from several arbitrary conventions. These results aid in application and algorithm development by determining the efficacy of a value set in computations.

翻译：GraphBLAS社区展示了线性代数赋能图算法的强大能力，例如通过矩阵向量积实现广度优先搜索（BFS）遍历。本文研究了在使用有向超图（由关联阵列表示，其元素取自带有二元加法与乘法运算的任意值集）进行此类计算时所需的代数条件。结果表明，一步BFS遍历等价于要求这些运算具备特定代数性质。在假设运算存在幺元0和1的前提下，我们证明这两个运算必须是无零和、无零因子，且0在乘法下必须是零化子。此外，结合律与交换律被证明是实现一步BFS计算独立于若干任意约定的充分必要条件。这些结果通过确定值集在计算中的有效性，为算法应用与开发提供了支持。