On the structure of loop-free non-negative edge-bipartite graphs

Abstract

We continue the study of a class of signed graphs called finite connected loop-free edge-bipartite graphs Δ (bigraphs, for short), started in Simson (2013) [33] and continued in Gąsiorek et al. (2016) [14] and Simson-Zając (2017) [38]. In this paper we present an algorithmic approach to the study of non-negative bigraphs Δ with n+r≥2 vertices of corank r≥1, that is, bigraphs with the symmetric Gram matrix GΔ:=12(GˇΔ+GˇΔtr)∈Mn+r(12Z) positive semi-definite of rank n≥1, where GˇΔ∈Mn+r(Z) is the non-symmetric Gram matrix of Δ. One of the main results of this paper are Theorems 2.1 and 2.17 asserting that every such a bigraph Δ with n+r≥2 vertices can be obtained from a corank zero (i.e. positive) connected loop-free bigraph Δ′ with n≥1 vertices by an r-point extension procedure (Δ′,u(1),…,u(r))↦Δ:=Δ′[[u(1),…,u(r)]] along the r≥1 roots u(1),…,u(r)∈Zn of the positive definite Gram form qΔ′:Zn→Z, v↦vGˇΔ′vtr. It is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop-free non-negative corank r≥1 bigraphs with n+r≥2 vertices, from the corank-zero bigraphs Δ′ with n≥1 vertices.