Abstract:
Abstract. Biased graphs abstract gain graphs, which are graphs whose oriented edges are labelled invertibly from a group. A biased, hence gain, graph has two natural matroids: the bias matroid G and the (complete) lift matroid L (or L 0). If the gain group is contained in the multiplicative group of a skew eld, the bias matroid is representable by vectors over and also in several ways by hyperplanes; two of these representations generalize the classical Ceva and Menelaos theorems, while one dualizes the vector representation. The vector representation is unique for `full ' but not for all gain graphs. The dualizing hyperplane representation can be abstracted away from elds to a form of equational logic. If the gain group is contained in the additive group of a skew eld, then the complete lift matroid is representable by vectors and dually by linear, projective, and ane hyperplanes. The representations are unique for the complete lift but not for the incomplete lift matroid.
Citations
|
128
|
Introduction to Geometry
– Coxeter
- 1989
|
|
79
|
Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes
– Zaslavsky
- 1975
|
|
42
|
Las Vergnas: Orientability of Matroids
– Bland, M
- 1978
|
|
39
|
Optimization Algorithms for Networks and Graphs
– Evans, Minieka
- 1992
|
|
37
|
Signed graphs
– Zaslavsky
- 1982
|
|
31
|
On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-radon partitions, and orientations of graphs
– Greene, Zaslavsky
- 1983
|
|
19
|
Uniquely representable combinatorial geometries
– Brylawski, Lucas
- 1976
|
|
18
|
The three matroids. III. Chromatic and dichromatic invariants. IV. Geometrical realizations
– Bias, balance, et al.
- 1989
|
|
17
|
Signed graph coloring
– Zaslavsky
- 1982
|
|
14
|
Vergnas, Convexity in oriented matroids
– Las
- 1980
|
|
6
|
A class of geometric lattices based on groups
– Dowling
- 1973
|
|
6
|
Bicircular geometry and the lattice of forests of a graph, Quart
– Zaslavsky
- 1982
|
|
5
|
Signed graphs, Discrete Appl
– Zaslavsky
- 1982
|
|
2
|
ik, Mehrdimensionales Analogon zu Satzen von Menelaos und
– Budinsky, Naden
- 1972
|
|
2
|
Graphentheoretische Methoden des Operations Research
– Hassig
- 1979
|
|
2
|
A certain generalization of the theorems of Menelaos and Ceva
– Klein
- 1973
|
|
2
|
Sur les activites des orientations d'une geometrie combinatoire
– Vergnas
- 1978
|
|
2
|
personal communication
– Sulzberger
- 1977
|
|
1
|
The theorems of Menelaos and Ceva in an n-dimensional ane space (in Romanian; French summary), An. Univ. Craiova Ser. a IV-a 1
– Boldescu
- 1970
|
|
1
|
Binary Spaces and Cutting Planes. CWI Tract 73. Centrum voor Wiskunde en Informatica
– Gerards, Graphs, et al.
- 1990
|
|
1
|
doctoral dissertation
– Slilaty
|
|
1
|
Perpendicular dissections of space, submitted
– Zaslavsky
|
|
