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 Delta, with m 2 >= 3 vertices (a class of signed graphs), started in Simson (2013) [49], by means of the non-symmetric Gram matrix G(Delta) is an element of Mm+2 (Z) of Delta, its symmetric Gram matrix G(Delta) := 1/2[G(Delta) + G(Delta)(tr)] is an element of Mm+2(1/2Z), the Gram quadratic form q(Delta), : Z(m+2) -> Z, and the Coxeter spectrum specc(Delta) subset of C, i.e., the complex spectrum of the Coxeter matrix Cox(Delta) := -G Delta . G(Delta)(-tr) is an element of 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 G Delta is an element of Mm+2 (Z) of Delta 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 Delta, with m + 2 >= 3 vertices, up to the weak Gram G Delta -congruence Delta similar to z Delta', where Delta similar to z Delta' means that G(Delta)' = B-tr . G Delta . B, for some B is an element of Mm+2(Z) such that det B = +/-1. By one-vertex extensions of the simily similar to laced Euclidean diagrams (A) over tilde (m), m >= 1, (D) over tilde (m), m >= 4, (E) over tilde (6),(E) over tilde (7),(E) over tilde (8), we construct a family of connected loop-free corank-two diagrams (A) over tilde ((2))(m), (E) over tilde ((2))(6), (E) over tilde ((2))(7), (E) over tilde ((2))(8) (called simply extended Euclidean diagrams) such that they classify all connected corank-two loop-free edge-bipartite graphs Delta, with m 2 >= 3 vertices, up to the weak Gram Z-congruence Delta similar to z Delta'. A structure of connected corank-two loop-free edge-bipartite graphs Delta is described. It is shown that every such Delta contains a connected positive edge-bipartite subgraph Delta', that is Z-congruent with a simply laced Dynkin diagram Dyn(Delta) (called the Dynkin type of Delta) such that Delta is a two-point extension Delta'[[u,w]] of Delta' along two roots u, w of the positive definite Gram form q(Delta)' : Z(m) -> Z. This yields a combinatorial algorithm (Delta', u, w) -> Delta'[[u, w]] allowing us to construct all connected corank-two loop-free edge-bipartite graphs Delta, with m + 2 >= 3 vertices and D = Dyn(Delta), from the triples (Delta', u, w), where Delta' is positive of the Dynkin type D, and u, w are roots of the positive definite Gram form q(Delta)' : Z(m) -> Z. (C) 2016 Elsevier Inc. All rights reserved.