Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C#
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...
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| Veröffentlicht in: | Algebra and Discrete Mathematics |
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| Datum: | 2013 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут прикладної математики і механіки НАН України
2013
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/152352 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and
 P-critical unit forms using Maple and C# / A. Polak, D. Simson // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 242–286. — Бібліогр.: 43 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862736118869393408 |
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| author | Polak, A. Simson, D. |
| author_facet | Polak, A. Simson, D. |
| citation_txt | Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and
 P-critical unit forms using Maple and C# / A. Polak, D. Simson // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 242–286. — Бібліогр.: 43 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | 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.
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| first_indexed | 2025-12-07T19:52:47Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-152352 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T19:52:47Z |
| publishDate | 2013 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Polak, A. Simson, D. 2019-06-10T11:09:15Z 2019-06-10T11:09:15Z 2013 Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and
 P-critical unit forms using Maple and C# / A. Polak, D. Simson // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 242–286. — Бібліогр.: 43 назв. — англ. 1726-3255 2010 MSC:15A63, 11Y16, 68W30, 05E10 16G20, 20B40, 15A21. https://nasplib.isofts.kiev.ua/handle/123456789/152352 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. The research is supported by Polish Research Grant NCN 2011/03/B/ST1/00824 en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C# Article published earlier |
| spellingShingle | 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. |
| title | Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C# |
| title_full | Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C# |
| title_fullStr | Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C# |
| title_full_unstemmed | Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C# |
| title_short | Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C# |
| title_sort | algorithms computing o(n, z)-orbits of p-critical edge-bipartite graphs and p-critical unit forms using maple and c# |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152352 |
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