کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6422755 | 1341217 | 2014 | 13 صفحه PDF | دانلود رایگان |
We develop algorithmic techniques for the Coxeter spectral analysis of the class UBigrn of connected loop-free positive edge-bipartite graphs Î with nâ¥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 Î by means of their Gram matrix GÌÎ, the (complex) spectrum speccÎâC of the Coxeter matrix CoxÎ:=âGÌÎâ GÌÎâtr, and the geometry of Weyl orbits in the set MorDÎ of matrix morsifications AâMn(Z) of a simply laced Dynkin diagram DÎâ{An,Dn,E6,E7,E8} associated with Î and mesh root systems of type DÎ. Our algorithms construct the Coxeter-Gram polynomials coxÎ(t)âZ[t] and mesh geometries of root orbits of small connected loop-free positive edge-bipartite graphs Î. We apply them to the study of the following Coxeter spectral analysis problem: Does the Z-congruence ÎâZÎâ²hold (i.e., the matrices GÌÎand GÌÎâ²are Z-congruent), for any pair of connected positive loop-free edge-bipartite graphs  Î,Îâ²in UBigrnsuch that speccÎ=speccÎâ²? The problem if any square integer matrix AâMn(Z) is Z-congruent with its transpose Atr is also discussed. We present a solution for graphs in UBigrn, with nâ¤6.
Journal: Journal of Computational and Applied Mathematics - Volume 259, Part B, 15 March 2014, Pages 815-827