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A Graph Theoretical Framework for the Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸 n

González Jiménez, Arturo Jesús

Fundamenta Informaticae

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2021

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Vol. 184, nr 1

49--82

EN

In the context of signed line graphs, this article introduces a modified inflation technique to study strong Gram congruence of non-negative (integral quadratic) unit forms, and uses it to show that weak and strong Gram congruence coincide among positive unit forms of Dynkin type An . The concept of inverse of a quiver is also introduced, and is used to obtain and analyze the Coxeter matrix of non-negative unit forms of Dynkin type An . With these tools, connected principal unit forms of Dynkin type An are also classified up to strong congruence.

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Inflation Agorithm for Cox-regular Postive Edge-bipartite Graphs with Loops

Makuracki B., Simson D., Zyglarski B.

Fundamenta Informaticae

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2017

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Vol. 153, nr 4

367--398

EN

We continue the study of finite connected edge-bipartite graphs Δ, with m ≥ 2 vertices (a class of signed graphs), started in [SIAM J. Discrete Math. 27(2013), 827-854] and developed in [Fund. Inform. 139(2015), 249-275, 145(2016), 19-48] by means of the non-symmetric Gram matrix ĞΔ ∊ Mn(Z) defining Δ, its symmetric Gram matrix GΔ:=1/2[ĞΔ+ĞtrΔ]∊ Mn(1/2Z), and the Gram quadratic form qΔ : Zn → Z. In the present paper we study connected positive Cox-regular edge-bipartite graphs Δ, with n ≥ 2 vertices, in the sense that the symmetric Gram matrix GΔ∊ Mn(Z) of Δ is positive definite. Our aim is to classify such Cox-regular edge-bipartite graphs with at least one loop by means of an inflation algorithm, up to the weak Gram Z-congruence Δ ~Z Δ', where Δ ~ZΔ' means that GΔ' = Btr.GΔ .B, for some B ∊ Mn(Z) such that det B = ±1. Our main result of the paper asserts that, given a positive connected Cox-regular edge-bipartite graph Δ with n ≥ 2 vertices and with at least one loop there exists a Cox-regular edge-bipartite Dynkin graph Dn ∊ {Bn, Cn, F4, G2} with loops and a suitably chosen sequence t-• of the inflation operators of one of the types Δ'↦t-aΔ' and Δ'↦t-abΔ' such that the composite operator Δ↦t-•Δ reduces Δ to the bigraph Dn such that Δ ~Z Dn and the bigraphs Δ, Dn have the same number of loops. The algorithm does not change loops and the number of vertices, and computes a matrix B ∊ Mn(Z), with det B = ±1, defining the weak Gram Z-congruence Δ ~Z Dn, that is, satisfying the equation GDn= Btr.GΔ.B.

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Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops [Part] 2. Application to Coxeter Spectral Analysis

Kasjan S., Simson D.

Fundamenta Informaticae

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2015

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Vol. 139, nr 2

185--209

EN

This is a second part of our two part paper with the same title. Following our Coxeter spectral study in [Fund. Inform. [123(2013), 447-490] and [SIAM J. Discr. Math. 27(2013), 827- 854] of the category UBigrn of loop-free edge-bipartite (signed) graphs Δ, with n = 2 vertices, we study here the larger category RBigrn of Cox-regular edge-bipartite graphs Δ (possibly with dotted loops), up to the usual Z-congruences ~Z and Z. The positive graphs Δ in RBigrn, with dotted loops, are studied by means of the complex Coxeter spectrum speccΔ C, the irreducible mesh root systems of Dynkin types Bn, n = 2, Cn, n = 3, F4, G2, the isotropy group Gl(n, Z)Δ (containing the Weyl group of Δ), and by applying the matrix morsification technique introduced in [J. Pure Appl. Algebra 215(2011), 13-24] and [Fund. Inform. [123(2013), 447-490]. One of our aims of our two part paper is to study the Coxeter spectral analysis question: "Does the congruence Δ Z Δ' hold, for any pair of connected positive graphs Δ,Δ' ∊ RBigrn such that speccΔ = speccΔ' and the numbers of loops in ΔandΔ' coincide?"We do it by a reduction to the Coxeter spectral study of the Gl(n, Z)D-orbits in the set MorD C Mn(Z) of matrix morsifications of a Dynkin diagram D = DΔ ∊ UBigrn associated with Δ. In this second part, we construct numeric algorithms for computing the connected positive edge-bipartite graphs Δ in RBigrn, for a fixed n = 2, mesh algorithms for computing the set of all Z-invertible matrices B ∊ Gl(n, Z) definining the Z-congruenceΔ Z Δ', for positive graphsΔ,Δ' ∊ RBigrn, with n geq2 fixed, and mesh-type algorithms for the mesh root systems Γ(R·Δ,ΦΔ). We also present a classification and a structure type results for positive Cox-regular edge-bipartite graphs Δ with dotted loops.

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A Horizontal Mesh Algorithm for a Class of Edge-bipartite Graphs and their Matrix Morsifications

Kaniecki M., Kosakowska J., Malicki P., Marczak G.

Fundamenta Informaticae

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2015

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Vol. 136, nr 4

345--379

EN

We construct a horizontal mesh algorithm for a study of a special type of mesh root systems of connected positive loop-free edge-bipartite graphs Δ, with n ≥ 2 vertices, in the sense of [SIAM J. Discrete Math. 27 (2013), 827–854] and [Fund. Inform. 124 (2013), 309-338]. Given such a loop-free edge-bipartite graph Δ, with the non-symmetric Gram matrix ˇGΔ ∈ Mn(Z) and the Coxeter transformation ΦA : Zn → Zn defined by a quasi-triangular matrix morsification A ∈ Mn(Z) of Δ satisfying a non-cycle condition, our combinatorial algorithm constructs a ΦA-mesh root system structure Γ(RΔ,ΦA) on the finite set of all ΦA-orbits of the irreducible root system RΔ := {v ∈ Zn; v · ˇGΔ · vtr = 1}. We apply the algorithm to a graphical construction of a ΦI - mesh root system structure Γ(RI ,ΦI ) on the finite set of ΦI -orbits of roots of any poset I with positive definite Tits quadratic form bqI : ZI → Z.

5

Toroidal Algorithms for Mesh Geometries of Root Orbits of the Dynkin Diagram D4

Simson D.

Fundamenta Informaticae

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2013

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Vol. 124, nr 3

339--364

EN

By applying symbolic and numerical computation and the spectral Coxeter analysis technique of matrix morsifications introduced in our previous paper [Fund. Inform. 124(2013)], we present a complete algorithmic classification of the rational morsifications and their mesh geometries of root orbits for the Dynkin diagram 4 The structure of the isotropy group Gl(4, {Z})D4 of D 4 is also studied. As a byproduct of our technique we show that, given a connected loop-free positive edge-bipartite graph Δ, with n ≥ 4 vertices (in the sense of our paper [SIAM J. Discrete Math. 27(2013)]) and the positive definite Gram unit formqΔ ; Zn→Z, any positive integer d ≥ 1 can be presented as d = qΔ(v), with v Є Zn In case n = 3, a positive integer d ≥ 1 can be presented as d = qΔ(v), with v Є Zn , if and only if d is not of the form 4a(16 · b + 14), where a and b are non-negative integers.

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A Framework for Coxeter Spectral Analysis of Edge-bipartite Graphs, their Rational Morsifications and Mesh Geometries of Root Orbits

Simson D.

Fundamenta Informaticae

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2013

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Vol. 124, nr 3

309--338

EN

Following the spectral Coxeter analysis of matrix morsifications for Dynkin diagrams, the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we continue our study of the category UBigrn of loop-free edge-bipartite (signed) graphs ∆, with n > 2 vertices, by means of the Coxeter number oa, the Coxeter spectrum specc∆ of ∆, that is, the spectrum of the Coxeter polynomial cox∆(t) ∈ Z[t] and the Z-bilinear Gram form b∆ : Zn x Zn →Z of ∆ [SIAM J. Discrete Math. 27(2013)]. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. We show that the Coxeter spectral classification of connected edge-bipartite graphs A in UBigrn reduces to the Coxeter spectral classification of rational matrix morsifications A ∈ MorD∆ for a simply-laced Dynkin diagram D∆ associated with ∆. Given ∆ in UBigrn, we study the isotropy subgroup Gl(n, Z)∆ of Gl(n, Z) that contains the Weyl group W∆. and acts on the set Mor∆ of rational matrix morsifications A of ∆ in such a way that the map A → (speccA, det A, c∆) is Gl(n, Z)∆-invariant. It is shown that, for n < 6, specc∆ is the spectrum of one of the Coxeter polynomials listed in Tables 3.11-3.11(a) (we determine them by computer search using symbolic and numeric computation). The question, if two connected positive edge-bipartite graphs ∆, ∆' in UBigrn, with specc∆= specc∆,, are Z-bilinear equivalent, is studied in the paper. The problem if any Z-invertible matrix A ∈ Mn(Z) is Z-congruent with its transpose Atr is also discussed.