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 Delta = (Delta(0), Delta(1)), i.e., for edge-bipartite graphs Delta, with n + 2 = vertical bar Delta(0)vertical bar >= 3 vertices such that its rational symmetric Gram matrix G(Delta) : = 1/2 (G(Delta) +G(Delta)(tr)) is an element of 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 Delta approximate to(Z) Delta', by means of the non-symmetric Gram matrix G(Delta) is an element of Mn+2 (Z), the Coxeter matrix Cox(Delta) := -G(Delta).G(Delta)(-tr) is an element of Mn+2 (Z), its complex spectrum specc(Delta) subset of C, and an associated simply laced Dynkin diagram Dyn(Delta) is an element of{A(n), D-n,D- E-6, E-7, E-8}, with n >= 1 vertices. it is known that if the congruence hold Delta approximate to(Z) Delta' (i.e., there exists B is an element of Mn+2 (Z) such that det B = +/- 1 and G(Delta) = B-tr . G(Delta) . B) the (specc(Delta), Dyn(Delta)) = (specc(Delta'), Dyn(Delta)'). The inverse implication is proved in [Fund. Inform. 152 (2017) 185-222] for all pairs of non-negative corank two bigraphs Delta, Delta' without loops with n + 2 <= 6 vertices. One of the main aims of the paper is to find algorithms that construct the set Cong(Delta, Delta') of all Z-invertible matrices B is an element of Mn+2(Z) defining the congruence Delta approximate to(Z) Delta', for any pair of corank two bigraphs Delta and Delta' such that (specc(Delta), Dyn(Delta)) =(specc(Delta)' similar to, Dyn(Delta)'). We do it in case when Delta and Delta' have at most six vertices in three steps: (i) to any such Delta, we associate algorithmically a Phi(Delta)-mesh translation quiver Gamma(Delta)center dot= Gamma((R) over cap (Delta), Phi(Delta)) in Z(n+2)(called Phi(Delta)-mesh geometry of roots of Delta, in the sense of Simson (2013) [42]), where Phi(Delta): Z(n+2) -> Z(n+2), v bar right arrow v center dot Cox(Delta), is the Coxeter transformation of Delta; (ii) we define an injective contravariant map (h) over tilde (center dot): Cong(Delta,Delta')-> Iso(Gamma(center dot)(Delta)', Gamma(center dot)(Delta)), where Iso(Gamma(center dot)(Delta), Gamma(center dot)(Delta)) is the set of all mesh translation quiver isomorphisms (h) over tilde (B): Gamma(center dot)(Delta) -> Gamma(center dot)(Delta), and (iii) a numerical algorithm computing a required matrix B is an element of Cong(Delta,Delta') is constructed by means of the shape of the Phi(Delta)-mesh translation quiver Gamma(center dot)(Delta), its toroidal-tubular structure, and an analysis of admissible isomorphisms (h) over tilde (B): Gamma(center dot)(Delta), -> Gamma(center dot)(Delta)G. (c) 2020 Elsevier Inc. All rights reserved