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

Bocian, Rafał; Felisiak, Mariusz; Simson, Daniel

doi:10.1016/j.cam.2013.07.013

Cited by 27 publications

(5 citation statements)

References 25 publications

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“…The reciprocal of Lemma 3 is not true in general. Thus, for instance, the two partial magmas ( [2], ·) and ( [2], •) that are respectively described by the non-zero products 1 · 1 = 1 and 1 • 1 = 1 = 2 • 1 are not isotopic. Nevertheless, the partial magma rings A · and A • , with respective bases {e 1 , e 2 } and {e ′ 1 , e ′ 2 }, are isotopic by means of the isotopism (f, Id, Id), where the linear transformation f is described by f (e 1 ) = e ′ 1 and f (e 2 ) = e ′ 2 − e ′ 1 .…”

Section: Isotopisms Of Algebrasmentioning

confidence: 99%

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Computation of isotopisms of algebras over finite fields by means of graph invariants

Falcón

Ganfornina

Núñez

et al. 2017

Journal of Computational and Applied Mathematics

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In this paper we define a pair of faithful functors that map isomorphic and isotopic finite-dimensional algebras over finite fields to isomorphic graphs. These functors reduce the cost of computation that is usually required to determine whether two algebras are isomorphic. In order to illustrate their efficiency, we determine explicitly the classification of two-and threedimensional partial quasigroup rings.

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Section: Isotopisms Of Algebrasmentioning

confidence: 99%

“…Graph invariants have been proposed in the last years as an efficient alternative to study isomorphisms of distinct types of algebras [2,4,15]. Nevertheless, the problem of identifying a functor that relates the category of algebras with that of graphs remains still open.…”

Section: Introductionmentioning

confidence: 99%

Computation of isotopisms of algebras over finite fields by means of graph invariants

Falcón

Ganfornina

Núñez

et al. 2017

Journal of Computational and Applied Mathematics

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“…Also we mention that the results and a technique used in the paper can be useful in the study of signed graphs in the sense of Harary [23] and Zaslavsky [53], and in the Coxeter spectral analysis of connected simply-laced edge-bipartite graphs recently developed in [37,40,41], see also [3,7], [30, Section 5], [32, Section 2.2], [25, 26, 27], [38, Section 5], and [39, 42].…”

Section: Ghasemi / An Algorithmic-type Classification Of Tetravalentmentioning

confidence: 99%

An Algorithmic-type Classification of Tetravalent One-regular Graphs Using Computer Algebra Tools

Ghasemi

2014

Fundamenta Informaticae

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A finite graph is one-regular if its automorphism group acts regularly on the set of its arcs. In the present paper, tetravalent one-regular graphs of order 7p 2 , where p is a prime, are classified using computer algebra tools. . One-regular Graphs Using Computer... is connected, and it is of valency 2 if and only if it is a cycle. In this sense the first non-trivial case is that of cubic graphs. The first example of a cubic one-regular graph was constructed by Frucht [14] and later on a lot of works have been done along this line (as part of the more general investigation of cubic arc-transitive graphs) see [9,10,11,12]. As we mentioned a graph with valency 4 is called tetravalent graph. Tetravalent one-regular graphs have also received considerable attention. In [5], tetravalent oneregular graphs of prime order were constructed. In [31], an infinite family of tetravalent one-regular Cayley graphs on alternating groups is given. Tetravalent one-regular circulant graphs were classified in [50] and tetravalent one-regular Cayley graphs on abelian groups were classified in [51]. Next, one may deduce from [29,43,45] a classification of tetravalent one-regular Cayley graphs on dihedral groups. Let p and q be primes. Then, clearly every tetravalent one-regular graph of order p is a circulant graph. Also, by [6,35,36,44,50, 51] every tetravalent one-regular graph of order pq or p 2 is a circulant graph. Furthermore, the classification of tetravalent one-regular graphs of order 3p 2 , 4p 2 , 5p 2 , 6p 2 and 2pq are given in [8,18,19,20,54]. Along this line we obtain Theorem 3.1, which is the main result of the paper. It contains a complete classification of tetravalent one-regular graphs of order 7p 2 , where p ≥ 2 is a prime. 212 M. Ghasemi / An Algorithmic-type Classification of TetravalentAlso we mention that the results and a technique used in the paper can be useful in the study of signed graphs in the sense of Harary [23] and Zaslavsky [53], and in the Coxeter spectral analysis of connected simply-laced edge-bipartite graphs recently developed in [37,40,41], see also [3,7], [30, Section 5], [32, Section 2.2], [25, 26, 27], [38, Section 5], and [39, 42].The proof of our classification result (Theorem 3.1) essentially uses an algorithmic and combinatorial reduction. In particular, we use a reduction to the set of arc-transitive graphs of order at most 640 and to recent results by Potočnik et al, obtained in [33,34]. In fact they used magma for getting these result.

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“…We recall from [3]- [13] and [24]- [27] that a motivation for the study of mesh root systems, mesh geometries of roots, matrix morsifications, and their connections with some Diophantine geometry problems and the Hilbert's tenth problem (see [18]) is given in [10] and [24]- [27]. We recall that root systems and Dynkin diagrams play an important role in several branches of mathematics including Lie theory, algebraic geometry, group theory, geometry, singularity theory, and representation theory, see [1], [27], [32].…”

Section: Introductionmentioning

confidence: 99%

On Coxeter Type Classification of Loop-Free Edge-Bipartite Graphs and Matrix Morsifications

Bocian

Felisiak

Simson

2013

2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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We continue and complete a Coxeter spectral study (presented in our talk given in SYNASC11 and SYNASC12) of the root systems in the sense of Bourbaki, the mesh geometries Γ(RΔ, ΦA) of roots of Δ in the sense of [J. Pure Appl. Algebra, 215 (2010), 13-34], and matrix morsifications A ∈ MorΔ, for simply laced Dynkin diagrams Δ ∈ {An, Dn, E6, E7, E8}. Here we report on algorithmic and morsification technique for the Coxeter spectral analysis of connected loop-free edge-bipartite graphs Δ with n ≥ 2 vertices by means of the Coxeter matrix CoxΔ ∈ Mn(Z), the Coxeter spectrum specc Δ , and an inflation algorithm associating to any connected loop-free positive bigraph Δ a simply laced Dynkin diagram DΔ, and defining a Z-congruence of the symmetric Gram matrices GΔ and GDΔ. We also present a computer aided technique that allows us to construct a Z-congruence of the non-symmetric Gram matricesǦΔ andǦ Δ , if the Coxeter spectra specc Δ and specc Δ coincide. A complete Coxeter spectral classification of positive edge-bipartite graphs Δ of Coxeter-Dynkin types DΔ ∈ {An, Dn, E6, E7}, with n ≤ 7, is obtained by a reduction to computer calculation of Gl(n, Z)DΔ-orbits in the set MorDΔ, where Gl(n, Z)DΔ is the isotropy group of the Dynkin diagram DΔ.

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

Cited by 27 publications

References 25 publications

Computation of isotopisms of algebras over finite fields by means of graph invariants

Computation of isotopisms of algebras over finite fields by means of graph invariants

An Algorithmic-type Classification of Tetravalent One-regular Graphs Using Computer Algebra Tools

On Coxeter Type Classification of Loop-Free Edge-Bipartite Graphs and Matrix Morsifications

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