Symbolic computation of strong Gram congruences for Cox-regular positive edge-bipartite graphs with loops

摘要

The paper can be viewed as a second part of the author's paper (Simson (2018) [43]). Our main aim is to construct symbolic algorithms for the Coxeter spectral classification of a class of signed graphs (called edge-bipartite graphs), with n >= 2 vertices, we started in Simson (2013) [37] and Bocian et al. (2014) [6]. More precisely, we construct algorithms for the classification (up to the strong Gram Z-congruence Delta approximate to(Z) Delta') of all finite connected Cox-regular edge-bipartite graphs Delta with at least one loop (bigraphs, for short) that are positive in the sense that the associated symmetric Gram matrix G(Delta) = 1/2 (G(Delta) + G(Delta)(tr)) is an element of M-n (1/2Z) is positive definite, where G(Delta) is an element of M-n(Z) is the non-symmetric Gram matrix of Delta defined in Section 1 and Delta approximate to(Z) Delta' means that there is a Z-invertible matrix B is an element of M-n(Z) such that G(Delta') = B-tr . G(Delta) . B. We recall from [43] that every such a bigraph Delta, with a loop, is strongly Gram Z-congruent with a bigraph D-Delta, that is one of the positive Cox-regular bigraphs B-n, n >= 2, C-n, n >= 3, F-4, M-4, G(2) presented in Section 1. Here, by applying the geometry of mesh root system technique, we construct symbolic algorithms that compute the correspondence Delta bar right arrow D-Delta and the set (Delta)G1(n, Z)(D Delta) of all Z-invertible matrices B is an element of M-n(Z) defining the strong Gram Z-congruence Delta approximate to(Z) D-Delta, for any connected positive bigraph Delta, with n >= 2 vertices and at least one loop. (C) 2019 Elsevier Inc. All rights reserved.