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 Delta = (Delta(0),Delta(1)). i.e. for edge-bipartite graphs Delta, with at least n - 3 vertices such that their rational symmetric Gram matrix G(Delta)is an element of M-n (Q) is positive semidefinite of rank n - 2. We study such connected edge-bipartite graphs by means of the nonsymmetric Gram matrix G(Delta) is an element of M-n (Z), the Coxeter matrix Cox(Delta) := -G(Delta)center dot G(Delta)(-tr) , its complex spectrum specc(Delta)subset of C, and an associated simply laced Dynkin diagram Dyn(Delta), with n - 2 vertices. Here Z means the ring of integers. It is well-known that if Delta approximate to(Z) Delta' (i.e., there exists B is an element of M-n (Z) such that det B = +/- 1 and G(Delta') = B-tr center dot G(Delta)center dot B) then specc(Delta)= specc(Delta') and Dyn(Delta) = Dyn(Delta'). A complete classification of connected non-negative loop-free edge-bipartite graphs Delta with at most six vertices of corank two, up to the Z -congruence Delta approximate to (Z)Delta', 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.