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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2013, Volume 16, Issue 2, Pages 242–286 (Mi adm451)

RESEARCH ARTICLE

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

A. Polak, D. Simson

Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

Abstract: 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 $n-1$ vertices and $w$ a sincere root of $\Delta$, up to an action $*:\mathcal{U} \mathcal{B} igr_n \times O(n,\mathbb{Z}) \to \mathcal{U}\mathcal{B} igr_n$ of the orthogonal group $O(n,\mathbb{Z})$ on the set $\mathcal{U} \mathcal{B} igr_n$ 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 $\mathcal{U} \mathcal{B} igr_n$ and for computing the $O(n, \mathbb{Z})$-orbits of $P$-critical graphs $\Delta$ in $\mathcal{U} \mathcal{B} igr_n$ 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 $\mathcal{U} \mathcal{B} igr_n$ and connected positive graphs in $\mathcal{U} \mathcal{B} igr_n$, together with their Coxeter polynomials, reduced Coxeter numbers, and the $O(n, \mathbb{Z})$-orbits, for $n\leq 10$. The computational results are presented in tables of Section 5.

Keywords: edge-bipartite graph, unit quadratic form, $P$-critical edge-bipartite graph, Gram matrix, sincere root, orthogonal group, algorithm, Coxeter polynomial, Euclidean diagram.

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Bibliographic databases:
MSC: 15A63, 11Y16, 68W30, 05E10 16G20, 20B40, 15A21
Revised: 26.07.2013
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Citation: A. Polak, D. Simson, “Algorithms computing $O(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C#”, Algebra Discrete Math., 16:2 (2013), 242–286

Citation in format AMSBIB
\Bibitem{PolSim13} \by A.~Polak, D.~Simson \paper Algorithms computing ${\rm O}(n, \mathbb{Z})$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\# \jour Algebra Discrete Math. \yr 2013 \vol 16 \issue 2 \pages 242--286 \mathnet{http://mi.mathnet.ru/adm451} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3186088} 

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This publication is cited in the following articles:
1. Marcin Gąsiorek, Daniel Simson, Katarzyna Zając, “Algorithmic computation of principal posets using Maple and Python”, Algebra Discrete Math., 17:1 (2014), 33–69
2. Polak A. Simson D., “Coxeter Spectral Classification of Almost Tp-Critical One-Peak Posets Using Symbolic and Numeric Computations”, Linear Alg. Appl., 445 (2014), 223–255
3. Polak A., Simson D., “Algorithmic Experiences in Coxeter Spectral Study of P-Critical Edge-Bipartite Graphs and Posets”, 2013 15Th International Symposium on Symbolic and Numeric Algorithms For Scientific Computing (Synasc 2013), eds. Bjorner N., Negru V., Ida T., Jebelean T., Petcu D., Watt S., Zaharie D., IEEE, 2014, 375–382
4. Gasiorek M., “Efficient Computation of the Isotropy Group of a Finite Graph: a Combinatorial Approach”, 2013 15Th International Symposium on Symbolic and Numeric Algorithms For Scientific Computing (Synasc 2013), International Symposium on Symbolic and Numeric Algorithms For Scientific Computing, ed. Bjorner N. Negru V. Ida T. Jebelean T. Petcu D. Watt S. Zaharie D., IEEE, 2014, 104–111
5. Gasiorek M. Simson D. Zajac K., “Structure and a Coxeter-Dynkin Type Classification of Corank Two Non-Negative Posets”, Linear Alg. Appl., 469 (2015), 76–113
6. Gasiorek M., Zajac K., “on Algorithmic Study of Non-Negative Posets of Corank At Most Two and Their Coxeter-Dynkin Types”, Fundam. Inform., 139:4 (2015), 347–367
7. M. Gasiorek, D. Simson, K. Zajac, “A Gram classification of non-negative corank-two loop-free edge-bipartite graphs”, Linear Alg. Appl., 500 (2016), 88–118
8. D. Simson, “Symbolic algorithms computing Gram congruences in the Coxeter spectral classification of edge-bipartite graphs, I. A Gram classification”, Fundam. Inform., 145:1 (2016), 19–48
9. D. Simson, “Symbolic algorithms computing Gram congruences in the Coxeter spectral classification of edge-bipartite graphs, II. Isotropy mini-groups”, Fundam. Inform., 145:1 (2016), 49–80
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