A Criterion for the Equivalence of Exponent Matrices

• Date: 2026
• Main Authors: Журавльов, Віктор, Зеленський, Олексій
• Format: Article
• Language: English
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Online Access: https://mcm-math.kpnu.edu.ua/article/view/360336
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

The paper establishes a criterion for the equivalence of reduced exponent matrices in terms of weighted admissible quivers. Exponent matrices arise naturally in the theory of tiled orders over discrete valuation rings and determine a number of structural properties of such orders, including their quivers. Therefore, the problem of recognizing when two reduced exponent matrices are equivalent is important both for the classification of algebraic objects and for the comparison of their associated graph models. The article proves that two reduced exponent matrices are equivalent if and only if their admissible quivers are isomorphic and the weights of the corresponding simple cycles coincide. This result strengthens the known necessary invariants of equivalence, such as equality of the sums of matrix entries and isomorphism of quivers, which are not sufficient by themselves. The criterion is formulated in the language of weight functions and cycle weights, which makes it convenient for use in combinatorial and graph-theoretic analysis of algebraic data. © Zhuravlev V. M., Zelenskiy O. V., 2026 From the viewpoint of mathematical modelling, an exponent matrix may be interpreted as an integer-valued directed distance model satisfying triangle-type constraints. In this interpretation, elementary transformations preserve cycle weights and correspond to gauge-type changes of vertex potentials. Thus the obtained criterion can be used to identify equivalent weighted directed graph models, to reduce redundant representations in algebraic modelling, and to compare discrete structures that arise in network models, optimization problems, representation-theoretic constructions, and other discrete models whose meaningful invariants are encoded by directed cycles. The result also gives a practical test for comparing representations without reconstructing the corresponding tiled orders.