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

• Published in: Fundamenta informaticae Vol. 152; no. 2; pp. 185 - 222
• Main Author: Zając, Katarzyna
• Format: Journal Article
• Language: English
• Published: London, England SAGE Publications 01.01.2017; Sage Publications Ltd
• Subjects: Binary system; Codes; Graph theory; Graphs; Integers; Mathematical analysis; Matrix methods; Dynkin diagram; Coxeter-Dynkin type; edge-bipartite graph; mesh geometry of roots
• ISSN: 0169-2968; 1875-8681
• DOI: 10.3233/FI-2017-1518

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 ∈ n (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 ∈ n (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 ∈ n (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.