http://dspace.nbuv.gov.ua:80/handle/123456789/150380 2026-04-18T23:56:23Z http://dspace.nbuv.gov.ua:80/handle/123456789/152356 Free n-nilpotent dimonoids Zhuchok, A.V. We construct a free n-nilpotent dimonoid and describe its structure. We also characterize the least n-nilpotent congruence on a free dimonoid, construct a new class of dimonoids with zero and give examples of nilpotent dimonoids of nilpotency index 2. 2013-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/152355 Classifying cubic s-regular graphs of orders 22p and 22p² Talebi, A.A.; Mehdipoor, N. A graph is s-regular if its automorphism group acts regularly on the set of s-arcs. In this study, we classify the connected cubic s-regular graphs of orders 22p and 22p² for each s ≥ 1, and each prime p. 2013-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/152353 Relative symmetric polynomials and money change problem Shahryari, M. This article is devoted to the number of non-negative solutions of the linear Diophantine equation a₁t₁ + a₂t₂ + ⋯ + antn = d, where a₁,…,an, and d are positive integers. We obtain a relation between the number of solutions of this equation and characters of the symmetric group, using relative symmetric polynomials. As an application, we give a necessary and sufficient condition for the space of the relative symmetric polynomials to be non-zero. 2013-01-01T00:00:00Z http://dspace.nbuv.gov.ua:80/handle/123456789/152352 Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C# Polak, A.; Simson, D. We present combinatorial algorithms constructing loop-free P-critical edge-bipartite (signed) graphs Δ′, with n ≥ 3 vertices, from pairs (Δ, w), with Δ a positive edge-bipartite graph having n-1 vertices and w a sincere root of Δ, up to an action ∗ : UBigrn × O(n, Z) → UBigrn of the orthogonal group O(n, Z) on the set UBigrn of loop-free edge-bipartite graphs, with n ≥ 3 vertices. Here Z is the ring of integers. We also present a package of algorithms for a Coxeter spectral analysis of graphs in UBigrn and for computing the O(n, Z)-orbits of P-critical graphs Δ in UBigrn as well as the positive ones. By applying the package, symbolic computations in Maple and numerical computations in C#, we compute P-critical graphs in UBigrn and connected positive graphs in UBigrn, together with their Coxeter polynomials, reduced Coxeter numbers, and the O(n, Z)-orbits, for n ≤ 10. The computational results are presented in tables of Section 5. 2013-01-01T00:00:00Z