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The GraphBLAS community has demonstrated the power of linearalgebra-leveraged graph algorithms, such as matrix-vector products forbreadth-first search (BFS) traversals. This paper investigates the algebraicconditions needed for such computations when working with directed hypergraphs,represented by incidence arrays with entries from an arbitrary value set withbinary addition and multiplication operations. Our results show the one-stepBFS traversal is equivalent to requiring specific algebraic properties of thoseoperations. Assuming identity elements 0, 1 for operations, we show that thetwo operations must be zero-sum-free, zero-divisor-free, and 0 must be anannihilator under multiplication. Additionally, associativity and commutativityare shown to be necessary and sufficient for independence of the one-step BFScomputation from several arbitrary conventions. These results aid inapplication and algorithm development by determining the efficacy of a valueset in computations.

Algebraic Conditions on One-Step Breadth-First Search