Affiliations: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland. [email protected]
Correspondence:
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Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland.
Abstract: 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.
Keywords: edge-bipartite graph, mesh geometry of roots, Dynkin diagram, Coxeter-Dynkin type
