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

• Date: 2018
• Main Authors: Polak, Agnieszka, Simson, Daniel
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2018
• Subjects: 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
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1161
• Journal Title: Algebra and Discrete Mathematics

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 \(\Delta \),...