Edge-bipartite Graphs Research Articles

Applications of mesh algorithms and self-dual mesh geometries of root Coxeter orbits to a Horn-Sergeichuk type problem

One of the main aims of the paper is to develop the mesh geometry technique for corank-two edge-bipartite graphs Δ with n+2≥3 vertices, and the mesh algorithms introduced in [30,33] and successfully studied in our recent article [42]. We introduce and study the concept of a self-duality of mesh geometries Γ(RˆΔ,ΦΔ) viewed as ΦΔ-mesh translation quivers. We show how self-dualities of mesh geometries Γ(RˆΔ,ΦΔ) and the mesh geometry technique is applied to an affirmative algorithmic solution of so called Horn-Sergeichuk type problem [9, Problem 4.3] on the self-congruency of square integer matrices A∈Mn+2(Z), for the class of non-symmetric Gram matrices A=GˇΔ of corank-two loop-free edge-bipartite graphs Δ, with n+2≤6 vertices. More precisely, we show that each of the mesh geometries Γ(RˆΔ,ΦΔ) is self-dual, we construct its dual form Γ(RˆΔ,ΦΔ)op=Γ(RˆΔ,ΦΔ−1) isomorphic with Γ(RˆΔ,ΦΔ), and we construct a canonical self-duality isomorphism fΔ:Γ(RˆΔ,ΦΔ)→Γ(RˆΔ,ΦΔ)op of mesh translation quivers. Using the self-duality fΔ we construct combinatorial algorithms such that, given a square Gram matrix A=GˇΔ∈Mn+2(Z) of Δ lying in this class, they are able to compute a Z-invertible matrix B∈Mn+2(Z) that coincide with its inverse B−1 and defines the congruence of A with Atr, i.e., the equation Atr=Btr⋅A⋅B is satisfied.An idea of our solution is outlined in Section 8 of our recent article [42], where among others two of our 13 algorithms solving the problem are constructed. The remaining 11 algorithms are constructed in the present article. We do it by means of the structure of the standard self-dual ΦΔ-mesh translation quiver Γ(RˆΔ,ΦΔ) (called a geometry) canonically associated with Δ, consisting of ΦΔ-meshes of ΦΔ-orbits OΔ(w) of vectors w∈RˆΔ⊆Zn+2, where ΦΔ:Zn+2→Zn+2 is the Coxeter transformation of Δ. We construct in the paper such self-dual ΦΔ-mesh geometry Γ(RˆΔ,ΦΔ), for each of the corank-two loop-free edge-bipartite graphs Δ, with n+2≤6 vertices.

On mesh geometries of root Coxeter orbits and mesh algorithms for corank two edge-bipartite signed graphs

Following a Coxeter spectral analysis problems for positive edge-bipartite graphs (signed multigraphs with a separation property) introduced in Simson (2013) [41] and Simson (2013) [42], we study analogous problems for loop-free corank two edge-bipartite graphs Δ=(Δ0,Δ1), i.e., for edge-bipartite graphs Δ, with n+2=|Δ0|≥3 vertices such that its rational symmetric Gram matrix GΔ:=12(GˇΔ+GˇΔtr)∈Mn+2(Q) is positive semi-definite of rank n. We study such connected edge-bipartite signed graphs (bigraphs, for short), up to the strong Gram congruence Δ≈ZΔ′, by means of the non-symmetric Gram matrix GˇΔ∈Mn+2(Z), the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Mn+2(Z), its complex spectrum speccΔ⊆C, and an associated simply laced Dynkin diagram DynΔ∈{An,Dn,E6,E7,E8}, with n≥1 vertices. It is known that if the congruence holds Δ≈ZΔ′ (i.e., there exists B∈Mn+2(Z) such that det⁡B=±1 and GˇΔ′=Btr⋅GˇΔ⋅B) then (speccΔ,DynΔ)=(speccΔ′,DynΔ′).The inverse implication is proved in [Fund. Inform. 152 (2017) 185–222] for all pairs of non-negative corank two bigraphs Δ, Δ′ without loops with n+2≤6 vertices. One of the main aims of the paper is to find algorithms that construct the set Cong(Δ,Δ′) of all Z-invertible matrices B∈Mn+2(Z) defining the congruence Δ≈ZΔ′, for any pair of corank two bigraphs Δ and Δ′ such that (speccΔ,DynΔ)=(speccΔ′,DynΔ′). We do it in case when Δ and Δ′ have at most six vertices in three steps: (i) to any such Δ, we associate algorithmically a ΦΔ-mesh translation quiver ΓΔ•=Γ(RˆΔ,ΦΔ) in Zn+2 (called ΦΔ-mesh geometry of roots of Δ, in the sense of Simson (2013) [42]), where ΦΔ:Zn+2→Zn+2, v↦v⋅CoxΔ, is the Coxeter transformation of Δ; (ii) we define an injective contravariant map h˜•:Cong(Δ,Δ′)→Iso(ΓΔ′•,ΓΔ•), where Iso(ΓΔ′•,ΓΔ•) is the set of all mesh translation quiver isomorphisms h˜B:ΓΔ′•→ΓΔ•, and (iii) a numerical algorithm computing a required matrix B∈Cong(Δ,Δ′) is constructed by means of the shape of the ΦΔ-mesh translation quiver ΓΔ•, its toroidal-tubular structure, and an analysis of admissible isomorphisms h˜B:ΓΔ′•→ΓΔ•.

On polynomial time inflation algorithm for loop-free non-negative edge-bipartite graphs

We study a class of signed graphs called finite connected loop-free edge-bipartite graphs Δ (bigraphs, for short), started in Simson (2013) and continued in Simson and Zając (2017) and Makuracki and Simson (2019). In this paper we present an algorithmic approach to the study of non-negative bigraphs Δ with n+r≥1 vertices Δ0={a1,…,an+r} of corank r≥0, that is, bigraphs with the symmetric Gram matrix GΔ≔12(GˇΔ+GˇΔtr)∈Mn+r(12Z) positive semi-definite of rank n≥1, where GˇΔ∈Mn+r(Z) is the non-symmetric Gram matrix of Δ.One of the main results of this paper asserts that every such a bigraph Δ with n+r≥1 vertices is algorithmically reduced by using at most O(n2) inflation operators tab−, a,b∈Δ0, to one of the canonical r-vertex extensions D̂n(r)∈{Ân(r),D̂n(r),Ê6(r),Ê7(r),Ê8(r)} of the simply laced Dynkin diagram Dn∈{An,Dn,E6,E7,E8}, where Dn is the Dynkin type of Δ, that is, a unique simply laced Dynkin graph Dn=DynΔ∈{An,Dn,E6,E7,E8} associated with Δ. We also construct inflation algorithm that: (a) computes in quadratic time a Dynkin type DynΔ of Δ, and (b) constructs a matrix B∈Mn+r(Z), with detB=±1, defining the weak Gram Z-congruence Δ∼ZBD̂n(r), that is, satisfying the equation GD̂n(r)=Btr⋅GΔ⋅B.

On the structure of loop-free non-negative edge-bipartite graphs

We continue the study of a class of signed graphs called finite connected loop-free edge-bipartite graphs Δ (bigraphs, for short), started in Simson (2013) [33] and continued in Gąsiorek et al. (2016) [14] and Simson-Zając (2017) [38]. In this paper we present an algorithmic approach to the study of non-negative bigraphs Δ with n+r≥2 vertices of corank r≥1, that is, bigraphs with the symmetric Gram matrix GΔ:=12(GˇΔ+GˇΔtr)∈Mn+r(12Z) positive semi-definite of rank n≥1, where GˇΔ∈Mn+r(Z) is the non-symmetric Gram matrix of Δ. One of the main results of this paper are Theorems 2.1 and 2.17 asserting that every such a bigraph Δ with n+r≥2 vertices can be obtained from a corank zero (i.e. positive) connected loop-free bigraph Δ′ with n≥1 vertices by an r-point extension procedure (Δ′,u(1),…,u(r))↦Δ:=Δ′[[u(1),…,u(r)]] along the r≥1 roots u(1),…,u(r)∈Zn of the positive definite Gram form qΔ′:Zn→Z, v↦vGˇΔ′vtr. It is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop-free non-negative corank r≥1 bigraphs with n+r≥2 vertices, from the corank-zero bigraphs Δ′ with n≥1 vertices.

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

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 Δ≈ZΔ′) of all finite connected Cox-regular edge-bipartite graphs Δ with at least one loop (bigraphs, for short) that are positive in the sense that the associated symmetric Gram matrix GΔ=12(GˇΔ+GˇΔtr)∈Mn(12Z) is positive definite, where GˇΔ∈Mn(Z) is the non-symmetric Gram matrix of Δ defined in Section 1 and Δ≈ZΔ′ means that there is a Z-invertible matrix B∈Mn(Z) such that GˇΔ′=Btr⋅GˇΔ⋅B. We recall from [43] that every such a bigraph Δ, with a loop, is strongly Gram Z-congruent with a bigraph DΔ, that is one of the positive Cox-regular bigraphs Bn, n≥2, Cn, n≥3, F4, M4, G2 presented in Section 1. Here, by applying the geometry of mesh root system technique, we construct symbolic algorithms that compute the correspondence Δ↦DΔ and the set GlˇΔ(n,Z)DΔ of all Z-invertible matrices B∈Mn(Z) defining the strong Gram Z-congruence Δ≈ZDΔ, for any connected positive bigraph Δ, with n≥2 vertices and at least one loop.

A Coxeter spectral classification of positive edge-bipartite graphs I. Dynkin types [formula omitted], [formula omitted], [formula omitted], [formula omitted], [formula omitted], [formula omitted], [formula omitted

We develop a computational technique for classification of a class of signed graphs (called edge-bipartite graphs), we started in Simson (2013) [42] and Bocian et al. (2014) [6]. Here we mainly study finite positive Cox-regular edge-bipartite graphs Δ (bigraphs, for short), with n≥2 vertices, by means of the non-symmetric Gram matrix GˇΔ∈Mn(Z) defining Δ, and the complex spectrum speccΔ⊂S1:={z∈C,|z|=1} of the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Mn(Z), called the Coxeter spectrum of Δ. The bigraph Δ is said to be positive if the symmetric Gram matrix GΔ=12(GˇΔ+GˇΔtr)∈Mn(Z) is positive definite.Our aim is to classify such edge-bipartite graphs, up to the strong Gram Z-congruence Δ≈ZΔ′, where Δ≈ZΔ′ means that GˇΔ′=Btr⋅GˇΔ⋅B, for some B∈Mn(Z) with det⁡B=±1. Our main result of the paper asserts that, given a pair Δ,Δ′ of Cox-regular connected positive edge-bipartite graphs with at least one loop, there is a congruence Δ≈ZΔ′ if and only if speccΔ=speccΔ′ and det⁡GˇΔ=det⁡GˇΔ′. Moreover, given n≥2, we present a list of five types of pairwise non-congruent bigraphs such that any Cox-regular connected positive bigraph with a loop and n≥2 vertices is strongly Z-congruent with a bigraph of the list. Our main idea used in the proof is a reduction of the classification problem to the problem of computing the orbits of a finite set MorSn⊆Mn(Z) of integer matrix morsifications of the antichain Sn consisting of n vertices, with respect to the right Gram action (A,B)↦A⁎B:=Btr⋅A⋅B of the integral orthogonal group O(n,Z) on MorSn. The computational technique developed in the paper allows also to construct a symbolic algorithm that computes a matrix B∈Gl(n,Z) defining the Gram Z-congruence Δ≈ZΔ′, if it does exist.

A Gram classification of principal Cox-regular edge-bipartite graphs via inflation algorithm

The paper is devoted to a Gram classification of an important class of signed graphs with loops playing an important role in many branches of mathematics, physics and computer science including game theory, crystallography, Diophantine geometry, Lie theory, spectral graph theory, a graph coloring technique, algebraic methods in graph theory. They have many deep applications in spectral machine learning methods, electrical networks, to link sign prediction in signed unipartite and bipartite networks, and community partition in social networks.We continue the study of finite connected edge-bipartite graphs Δ (bigraphs, for short), with m≥2 vertices (a class of signed graphs), started in Simson (2013) and developed in Simson (2016) by means of the non-symmetric Gram matrix GˇΔ∈Mm(Z) defining Δ, its symmetric Gram matrix GΔ:=12[GˇΔ+GˇΔtr]∈Mm(12Z), and the Gram quadratic form qΔ:Zm→Z. In the present paper, given n≥1, we mainly study connected principal Cox-regular edge-bipartite graphs Δ with loops and n+1≥2 vertices, in the sense that the symmetric Gram matrix GΔ∈Mn+1(Z) of Δ is positive semi-definite of rank n≥1. Our aim is to classify such edge-bipartite graphs by means of an inflation algorithm, up to the weak Gram Z-congruence Δ∼ZΔ′, where Δ∼ZΔ′ means that GΔ′=Btr⋅GΔ⋅B, for some B∈Mn+1(Z) with detB=±1. Our main result of the paper asserts that, given a principal Cox-regular connected edge-bipartite graph Δ with n+1≥2 vertices, there exists an edge-bipartite Euclidean graph Dn+1 of Tables A and B, and a suitably chosen sequence t•− of the inflation operators, each of one of the forms Δ′↦ta−Δ′ or Δ′↦tab−Δ′, such that the composite operator Δ↦t•−Δ reduces Δ to the bigraph Dn+1 such that: (i) Δ∼ZDn+1 and (ii) the bigraphs Δ, and Dn+1 have the same number of loops. The algorithm does not change loops and computes a matrix B∈Mn+1(Z), with detB=±1, defining the weak Gram Z-congruence Δ∼ZDn+1, that is, satisfying the equation GDn+1=Btr⋅GΔ⋅B.

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

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

Numeric Algorithms for Corank Two Edge-bipartite Graphs and their Mesh Geometries of Roots

Following a Coxeter spectral analysis problems for positive edge-bipartite graphs (signed multigraphs with a separation property) introduced in [SIAM J. Discr. Math. 27(2013), 827-854] and [Fund. Inform. 123(2013), 447-490], we study analogous problems for loop-free corank two edge-bipartite graphs Δ = (Δ0,Δ1). i.e. for edge-bipartite graphs Δ, with at least n = 3 vertices such that their rational symmetric Gram matrix GΔ ∈ 𝕄n(ℚ) is positive semi-definite of rank n – 2. We study such connected edge-bipartite graphs by means of the non-symmetric Gram matrix ĞΔ ∈ 𝕄n(ℤ), the Coxeter matrix CoxΔ := –ĞΔ · ĞΔ–tr, its complex spectrum speccΔ ⊆ ℂ, and an associated simply laced Dynkin diagram DynΔ, with n – 2 vertices. Here ℤ means the ring of integers. It is well-known that if Δ ≈ℤ Δ′ (i.e., there exists B ∈ 𝕄n(ℤ) such that det B = ±1 and ĞΔ′ = Btr · ĞΔ · B) then speccΔ = speccΔ′ and DynΔ = DynΔ′. A complete classification of connected non-negative loop-free edge-bipartite graphs Δ with at most six vertices of corank two, up to the ℤ-congruence Δ ≈ℤ Δ′, is also given. A complete list of representatives of the ℤ-congruence classes of all connected non-negative edge-bipartite graphs of corank two with with at most 6 vertices is constructed; it consists of 1, 2, 2 and 8 edge-bipartite graphs of corank two with 3, 4, 5 and 6 vertices, respectively.

Inflation algorithm for loop-free non-negative edge-bipartite graphs of corank at least two

We continue the study of finite connected loop-free edge-bipartite graphs Δ, with m≥3 vertices (a class of signed graphs), we started in Simson (2013) [48] and M. Gąsiorek et al. (2016) [19] by means of the non-symmetric Gram matrix GˇΔ∈Mm(Z) of Δ, its symmetric Gram matrix GΔ:=12[GˇΔ+GˇΔtr]∈Mm(12Z), and the Gram quadratic form qΔ:Zm→Z. In particular, we study connected non-negative edge-bipartite graphs Δ, with n+r≥3 vertices of corank r≥2, in the sense that the symmetric Gram matrix GΔ∈Mn+r(Z) of Δ is positive semi-definite of rank n≥1. The edge-bipartite graphs Δ of corank r≥2 are studied, up to the weak Gram Z-congruence Δ∼ZΔ′, where Δ∼ZΔ′ means that GΔ′=Btr⋅GΔ⋅B, for some B∈Mn+r(Z) such that det⁡B=±1. Our main result of the paper asserts that, given a connected edge-bipartite graph Δ with n+r≥3 vertices of corank r≥2, there exists a suitably chosen sequence t⁎− of the inflation operators of the form Δ′↦tab−Δ′ such that the composite operator Δ↦t⁎−Δ reduces Δ to a connected bigraph Dˆn(r) such that Δ∼ZDˆn(r) and Dˆn(r) is one of the canonical r-vertex extensions Aˆn(r), n≥1, Dˆn(r), n≥4, Eˆ6(r), Eˆ7(r), and Eˆ8(r), with n+r vertices, of the simply laced Dynkin diagrams An,Dn,E6,E7,E8, with n≥1 vertices. The algorithm constructs also a matrix B∈Mn+r(Z), with det⁡B=±1, defining the weak Gram Z-congruence Δ∼ZDˆn(r), that is, satisfying the equation GDˆn(r)=Btr⋅GΔ⋅B.

Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition I. Inflation Algorithm Revisited*

Congruences of Edge-bipartite Graphs with Applications to Grothendieck Group Recognition I. Inflation Algorithm Revisited*

Symbolic Algorithms Computing Gram Congruences in the Coxeter Spectral Classification of Edge-bipartite Graphs, I. A Gram Classification

We continue the Coxeter spectral study of the category UBigrm of loop-free edgebipartite (signed) graphs ∆, with m ≥ 2 vertices, we started in [SIAM J. Discr. Math. 27(2013), 827-854] for corank r = 0 and r = 1. Here we study the class of all non-negative edge-bipartite graphs ∆ ∈ UBigrn+r of corank r ≥ 0, up to a pair of the Gram Z-congruences ∼Z and ≈Z, by means of the non-symmetric Gram matrix Ǧ∆ ∈ Mn+r(Z) of ∆, the symmetric Gram matrix G∆ := 1 2 [Ǧ∆ + Ǧ tr ∆ ] ∈ Mn+r(Z), the Coxeter matrix Cox∆ := −Ǧ∆ · Ǧ −tr ∆ ∈ Mn+r(Z) and its spectrum specc∆ ⊂ C, called the Coxeter spectrum of ∆. One of the aims in the study of the category UBigrn+r is to classify the equivalence classes of the non-negative edge-bipartite graphs in UBigrn+r with respect to each of the Gram congruences ∼Z and ≈Z. In particular, the Coxeter spectral analysis question, when the strong congruence ∆ ≈Z ∆′ holds (hence also ∆ ∼Z ∆′ holds), for a pair of connected non-negative graphs ∆,∆′ ∈ UBigrn+r such that specc∆ = specc∆′ , is studied in the paper. One of our main aims is an algorithmic description of a matrix B defining the Gram Z-congruences ∆ ≈Z ∆′ and ∆ ∼Z ∆′, that is, a Z-invertible matrix B ∈Mn+r(Z) such that Ǧ∆′ = B · Ǧ∆ ·B and G∆′ = B ·G∆ ·B, respectively. We show that, given a connected non-negative edge-bipartite graph ∆ in UBigrn+r of corank r ≥ 0 there exists a simply laced Dynkin diagram D, with n vertices, and a connected canonical r-vertex extension D := D of D of corank r (constructed in Section 2) such that ∆ ∼Z D. We also show that every matrix B defining the strong Gram Zcongruence ∆ ≈Z ∆′ in UBigrn+r has the form B = C∆ · B · C−1 ∆′ , where C∆, C∆′ ∈ Mn+r(Z) are fixed Z-invertible matrices defining the weak Gram congruences ∆ ∼Z D and ∆′ ∼Z D with an r-vertex extended graph D, respectively, and B ∈ Mn+r(Z) is Z-invertible matrix lying in the isotropy group Gl(n+r,Z)D of D. Moreover, each of the columns κ ∈ Z n+r of B is a root of ∆, Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Torun, Poland. ∗Supported by Polish Research Grant NCN 2011/03/B/ST1/00824. Received November 2015; revised March 2016 20 D. Simson / Symbolic Algorithms Computing the Gram Congruence, I i.e., κ · Ǧ∆ · κ = 1. Algorithms constructing the set of all such matrices B are presented in case when r = 0. We essentially use our construction of a morsification reduction map φD : UBigrD → MorD that reduces (up to ≈Z) the study of the set UBigrD of all connected non-negative edgebipartite graphs ∆ in UBigrn+r such that ∆ ∼Z D to the study of Gl(n+r,Z)D-orbits in the set MorD ⊆ Gl(n+r,Z) of all matrix morsifications of the graph D.

A Gram classification of non-negative corank-two loop-free edge-bipartite graphs

We continue the Coxeter spectral study of finite connected loop-free edge-bipartite graphs Δ, with m+2≥3 vertices (a class of signed graphs), started in Simson (2013) [49], by means of the non-symmetric Gram matrix GˇΔ∈Mm+2(Z) of Δ, its symmetric Gram matrix GΔ:=12[GˇΔ+GˇΔtr]∈Mm+2(12Z), the Gram quadratic form qΔ:Zm+2→Z, and the Coxeter spectrum speccΔ⊂C, i.e., the complex spectrum of the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Gl(m+2,Z). In the present paper we study non-negative edge-bipartite graphs of corank two, in the sense that the symmetric Gram matrix GΔ∈Mm+2(Z) of Δ is positive semi-definite of rank m≥1. One of our aims is to get a complete classification of all connected corank-two loop-free edge-bipartite graphs Δ, with m+2≥3 vertices, up to the weak Gram Z-congruence Δ∼ZΔ′, where Δ∼ZΔ′ means that GΔ′=Btr⋅GΔ⋅B, for some B∈Mm+2(Z) such that det⁡B=±1. By one-vertex extensions of the simply laced Euclidean diagrams A˜m, m≥1, D˜m, m≥4, E˜6,E˜7,E˜8, we construct a family of connected loop-free corank-two diagrams A˜m(2),D˜m(2),E˜6(2),E˜7(2),E˜8(2) (called simply extended Euclidean diagrams) such that they classify all connected corank-two loop-free edge-bipartite graphs Δ, with m+2≥3 vertices, up to the weak Gram Z-congruence Δ∼ZΔ′. A structure of connected corank-two loop-free edge-bipartite graphs Δ is described. It is shown that every such Δ contains a connected positive edge-bipartite subgraph Δ′, that is Z-congruent with a simply laced Dynkin diagram DynΔ (called the Dynkin type of Δ) such that Δ is a two-point extension Δ′[[u,w]] of Δ′ along two roots u,w of the positive definite Gram form qΔ′:Zm→Z. This yields a combinatorial algorithm (Δ′,u,w)↦Δ′[[u,w]] allowing us to construct all connected corank-two loop-free edge-bipartite graphs Δ, with m+2≥3 vertices and D=DynΔ, from the triples (Δ′,u,w), where Δ′ is positive of the Dynkin type D, and u,w are roots of the positive definite Gram form qΔ′:Zm→Z.

Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops, I. Mesh Root Systems

Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops, I. Mesh Root Systems

Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops, II. Application to Coxeter Spectral Analysis

Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops, II. Application to Coxeter Spectral Analysis

Applications of matrix morsifications to Coxeter spectral study of loop-free edge-bipartite graphs

Following the spectral graph theory and a Coxeter type technique applied in the representation theory of groups, algebras, coalgebras, and posets, we study the category UBigrn of connected loop-free edge-bipartite graphs Δ with n≥2 vertices (a class of signed graphs) by means of their Coxeter spectrum speccΔ⊂C, i.e., the spectrum of the Coxeter matrix CoxΔ:=−ǦΔ⋅ǦΔ∈Mn(Z)(1.2), where ǦΔ∈Mn(Z) is the non-symmetric Gram matrix (1.1) of Δ. One of our aims is to compute the set CGpoln+ of all polynomials coxΔ(t), with positive graphs Δ in UBigrn, for all n≥2. Moreover, we study the following Coxeter spectral analysis problem (1.4): “Does the Z-congruence Δ≈ZΔ′ hold (i.e. the matrices ǦΔ and ǦΔ′ are Z-congruent), for any pair of connected positive loop-free edge-bipartite graphs Δ, Δ′ such that speccΔ=speccΔ′”. We show that our problems reduce to the study of the set MorD⊆Mn(Z) of matrix morsifications and their mesh root systems, for the simply laced Dynkin diagrams An, Dn, E6, E7, E8. On this way we construct algorithms allowing us to compute the finite set CGpoln+ and we get a solution of the problem (1.4) for n≤9 and Δ,Δ′ of types An and Dn.

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

We develop algorithmic techniques for the Coxeter spectral analysis of the class UBigrn of connected loop-free positive edge-bipartite graphs Δ 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 Δ by means of their Gram matrix ǦΔ, the (complex) spectrum speccΔ⊆C of the Coxeter matrix CoxΔ:=−ǦΔ⋅ǦΔ−tr, and the geometry of Weyl orbits in the set MorDΔ of matrix morsifications A∈Mn(Z) of a simply laced Dynkin diagram DΔ∈{An,Dn,E6,E7,E8} associated with Δ and mesh root systems of type DΔ. Our algorithms construct the Coxeter–Gram polynomials coxΔ(t)∈Z[t] and mesh geometries of root orbits of small connected loop-free positive edge-bipartite graphs Δ. We apply them to the study of the following Coxeter spectral analysis problem: Does theZ-congruenceΔ≈ZΔ′hold (i.e., the matricesǦΔandǦΔ′areZ-congruent), for any pair of connected positive loop-free edge-bipartite graphs Δ,Δ′inUBigrnsuch thatspeccΔ=speccΔ′? The problem if any square integer matrix A∈Mn(Z) is Z-congruent with its transpose Atr is also discussed. We present a solution for graphs in UBigrn, with n≤6.

An inflation algorithm and a toroidal mesh algorithm for edge-bipartite graphs

Abstract We present algorithmic methods used in the spectral analysis of bigraphs Δ by means of the Coxeter matrix Cox Δ , the Coxeter spectrum specc Δ , and an inflation algorithm associating to any connected loop-free positive bigraph Δ a simply-laced Dynkin diagram D Δ, and defining a Z -congruence of the symmetric Gram matrices G Δ and G D Δ . We also present a toroidal mesh algorithm that allows us to construct a Z -congruence of the non-symmetric Gram matrices G ˇ Δ and G ˇ Δ ′ , if the Coxeter spectra specc Δ and specc Δ ′ coincide.

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

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 Δ, with n ≥ 2 vertices, by means of the Coxeter number cΔ, the Coxeter spectrum speccΔ of Δ, that is, the spectrum of the Coxeter polynomial coxΔt ∈ $\mathbb{Z}$[t] and the $\mathbb{Z}$-bilinear Gram form bΔ : $\mathbb{Z}$n × $\mathbb{Z}$n → $\mathbb{Z}$ of Δ [SIAM J. Discrete Math. 272013]. 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 Δ in 𝒰Bigrn reduces to the Coxeter spectral classification of rational matrix morsifications A ∈ $\widehat{M}$orDΔ for a simply-laced Dynkin diagram DΔ associated with Δ. Given Δ in 𝒰Bigrn, we study the isotropy subgroup Gln,$\mathbb{Z}$Δ of Gln, $\mathbb{Z}$ that contains the Weyl group $\mathbb{W}$Δ and acts on the set $\widehat{M}$orΔ of rational matrix morsifications A of Δ in such a way that the map A $\mapsto$ speccA, det A, cΔ is Gln, $\mathbb{Z}$Δ-invariant. It is shown that, for n ≤ 6, speccΔ is the spectrum of one of the Coxeter polynomials listed in Tables 3.11-3.11a we determine them by computer search using symbolic and numeric computation. The question, if two connected positive edge-bipartite graphs Δ,Δ' in 𝒰Bigrn, with speccΔ = speccΔ', are $\mathbb{Z}$-bilinear equivalent, is studied in the paper. The problem if any $\mathbb{Z}$-invertible matrix A ∈ Mn$\mathbb{Z}$ is $\mathbb{Z}$-congruent with its transpose Atr is also discussed.

Inflation Algorithms for Positive and Principal Edge-bipartite Graphs and Unit Quadratic Forms

We describe combinatorial algorithms that compute the Dynkin type (resp. Euclidean type) of any positive (resp. principal) unit quadratic form q : $\mathbb{N}$n → $\mathbb{N}$ and of any positive (resp. principal) edge-bipartite connected graph Δ. The study of the problem is inspired by applications of the algorithms in the representation theory, in solving a class of Diophantine equations, in the study of mesh geometries of roots, in the spectral analysis of graphs, and in the Coxeter-Gram classification of edge-bipartite graphs.