Biased graphs IV: Geometrical realizations

Biased graphs IV: Geometrical realizations

A gain graph is a graph whose oriented edges are labelled invertibly from a group G, the gain group. A gain graph determines a biased graph and therefore has three natural matroids (as shown in Parts I–II): the bias matroid G has connected circuits; the complete lift matroid L0 and its restriction to the edge set, the lift matroid L, have circuits not necessarily connected. We investigate repre...

Biased Graphs . Iv

Geometrical Realizations of Shadow Geometries

Contents Introduction 615 General definitions and notation 618 § 1. Shadow geometries of incidence geometries 619 § 2. Shadow geometries of chamber systems and their geometrical reali-zations 626 § 3. Interpreting the shadow geometries as tessellations. .. . 6 3 1 § 4. Transitive tessellations for reflection groups 636 § 5. The Delaney symbols of the tessellations with a transitive reflection g...

Bounding and stabilizing realizations of biased graphs with a fixed group

Given a group Γ and a biased graph (G,B), we define a what is meant by a Γ-realization of (G,B) and a notion of equivalence of Γ-realizations. We prove that for a finite group Γ and t ≥ 3, that there are numbers n(Γ) and n(Γ, t) such that the number of Γ-realizations of a vertically 3-connected biased graph is at most n(Γ) and that the number of Γ-realizations of a nonseparable biased graph wit...

Biased Graphs