Numeric and mesh algorithms for the Coxeter spectral study of positive edge-bipartite graphs and their isotropy groups

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:259

Article

Bocian, Rafal¹  Felisiak, Mariusz¹  Simson, Daniel¹ 

[1] Nicolaus Copernicus Univ, Fac Math & Comp Sci, PL-87100 Torun, Poland

关键词: Edge-bipartite graph;    Matrix morsification;    Dynkin diagram;    Coxeter polynomial;    Mesh geometry of roots;    Computer algorithm;   

DOI  :  10.1016/j.cam.2013.07.013

来源: Elsevier

【 摘 要 】

We develop algorithmic techniques for the Coxeter spectral analysis of the class UBigr(n) of connected loop-free positive edge-bipartite graphs Delta with n >= 2 vertices (i.e., signed graphs). In particular, we present numerical and graphical algorithms allowing us a computer search in the study of such graphs Delta by means of their Gram matrix G(Delta), the (complex) spectrum specc(Delta) subset of C of the Coxeter matrix Cox(Delta) := -G(Delta) . G(Delta)(-tr) and the geometry of Weyl orbits in the set Mor(D Delta) of matrix morsifications A is an element of M-n(Z) of a simply laced Dynkin diagram D Delta is an element of {A(n), D-n, E-6, E-7, E-8} associated with Delta and mesh root systems of type D Delta. Our algorithms construct the Coxeter-Gram polynomials cox(Delta)(t) is an element of 7 vertical bar t vertical bar and mesh geometries of root orbits of small connected loop-free positive edge-bipartite graphs Delta. We apply them to the study of the following Coxeter spectral analysis problem: Does the Z-congruence Delta approximate to(Z) Delta' hold (i.e., the matrices G(Delta), and G(Delta'), are Z-congruent), for any pair of connected positive loop-free edge-bipartite graphs Delta, Delta' in UBigr(n), such that specc(Delta) = specc(Delta')? The problem if any square integer matrix A is an element of M-n(Z) is Z-congruent with its transpose A(tr) is also discussed. We present a solution for graphs in UBigr(n), with n <= 6. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

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