$Algorithms computing ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\#$

Algorithms computing ${\rm O}(n, \mathbb{Z})$-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 $\Delta'$, with $n\geq 3$ vertices, from pairs $(\Delta , w)$, with $\Delta $ a positive edge-bipartite graph having $n\mbox{-}1$ vertices and $w$ a sincere root of $\Delta $,...

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Дата:	2018
Автори:	Polak, Agnieszka, Simson, Daniel
Формат:	Стаття
Мова:	English
Опубліковано:	Lugansk National Taras Shevchenko University 2018
Теми:	edge-bipartite graph unit quadratic form $P$-critical edge-bipartite graph Gram matrix sincere root orthogonal group algorithm Coxeter polynomial Euclidean diagram 15A63 11Y16 68W30 05E10 16G20 20B40 15A21
Онлайн доступ:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1161
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Algebra and Discrete Mathematics

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Algebra and Discrete Mathematics

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spelling	oai:ojs.admjournal.luguniv.edu.ua:article-11612018-05-16T05:04:06Z Algorithms computing ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\# Polak, Agnieszka Simson, Daniel edge-bipartite graph, unit quadratic form, $P$-critical edge-bipartite graph, Gram matrix, sincere root, orthogonal group, algorithm, Coxeter polynomial, Euclidean diagram 15A63, 11Y16, 68W30, 05E10 16G20, 20B40, 15A21 We present combinatorial algorithms constructing loop-free $P$-critical edge-bipartite (signed) graphs $\Delta'$, with $n\geq 3$ vertices, from pairs $(\Delta , w)$, with $\Delta $ a positive edge-bipartite graph having $n\mbox{-}1$ vertices and $w$ a sincere root of $\Delta $, up to an action $*:\cal U \cal B igr_n \times {\rm O}(n,\mathbb{Z}) \to \cal U \cal B igr_n$ of the orthogonal group ${\rm O}(n,\mathbb{Z})$ on the set $\cal U \cal B igr_n$ of loop-free edge-bipartite graphs, with $n\geq 3$ vertices. Here $\mathbb{Z}$ is the ring of integers. We also present a package of algorithms for a Coxeter spectral analysis of graphs in $\cal U \cal B igr_n$ and for computing the ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical graphs $\Delta$ in $\cal U \cal B igr_n$ 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 $\cal U \cal B igr_n$ and connected positive graphs in $\cal U \cal B igr_n$, together with their Coxeter polynomials, reduced Coxeter numbers, and the ${\rm O}(n, \mathbb{Z})$-orbits, for $n\leq 10$. The computational results are presented in tables of Section 5. Lugansk National Taras Shevchenko University 2018-05-16 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1161 Algebra and Discrete Mathematics; Vol 16, No 2 (2013) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1161/653 Copyright (c) 2018 Algebra and Discrete Mathematics
institution	Algebra and Discrete Mathematics
baseUrl_str
datestamp_date	2018-05-16T05:04:06Z
collection	OJS
language	English
topic	edge-bipartite graph unit quadratic form $P$-critical edge-bipartite graph Gram matrix sincere root orthogonal group algorithm Coxeter polynomial Euclidean diagram 15A63 11Y16 68W30 05E10 16G20 20B40 15A21
spellingShingle	edge-bipartite graph unit quadratic form $P$-critical edge-bipartite graph Gram matrix sincere root orthogonal group algorithm Coxeter polynomial Euclidean diagram 15A63 11Y16 68W30 05E10 16G20 20B40 15A21 Polak, Agnieszka Simson, Daniel Algorithms computing ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\#
topic_facet	edge-bipartite graph unit quadratic form $P$-critical edge-bipartite graph Gram matrix sincere root orthogonal group algorithm Coxeter polynomial Euclidean diagram 15A63 11Y16 68W30 05E10 16G20 20B40 15A21
format	Article
author	Polak, Agnieszka Simson, Daniel
author_facet	Polak, Agnieszka Simson, Daniel
author_sort	Polak, Agnieszka
title	Algorithms computing ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\#
title_short	Algorithms computing ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\#
title_full	Algorithms computing ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\#
title_fullStr	Algorithms computing ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\#
title_full_unstemmed	Algorithms computing ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\#
title_sort	algorithms computing ${\rm o}(n, \mathbb{z})$-orbits of $p$-critical edge-bipartite graphs and $p$-critical unit forms using maple and c\#
description	We present combinatorial algorithms constructing loop-free $P$-critical edge-bipartite (signed) graphs $\Delta'$, with $n\geq 3$ vertices, from pairs $(\Delta , w)$, with $\Delta $ a positive edge-bipartite graph having $n\mbox{-}1$ vertices and $w$ a sincere root of $\Delta $, up to an action $*:\cal U \cal B igr_n \times {\rm O}(n,\mathbb{Z}) \to \cal U \cal B igr_n$ of the orthogonal group ${\rm O}(n,\mathbb{Z})$ on the set $\cal U \cal B igr_n$ of loop-free edge-bipartite graphs, with $n\geq 3$ vertices. Here $\mathbb{Z}$ is the ring of integers. We also present a package of algorithms for a Coxeter spectral analysis of graphs in $\cal U \cal B igr_n$ and for computing the ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical graphs $\Delta$ in $\cal U \cal B igr_n$ 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 $\cal U \cal B igr_n$ and connected positive graphs in $\cal U \cal B igr_n$, together with their Coxeter polynomials, reduced Coxeter numbers, and the ${\rm O}(n, \mathbb{Z})$-orbits, for $n\leq 10$. The computational results are presented in tables of Section 5.
publisher	Lugansk National Taras Shevchenko University
publishDate	2018
url	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1161
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first_indexed	2025-07-17T10:35:04Z
last_indexed	2025-07-17T10:35:04Z
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Algorithms computing \({\rm O}(n, \mathbb{Z})\)-orbits of \(P\)-critical edge-bipartite graphs and \(P\)-critical unit forms using Maple and C\#

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