Biased graphs. I: Bias, balance, and gains
type: Item
instance of: scholarly article
title: Biased graphs. I: Bias, balance, and gains
zbMATH Open document ID: 0714.05057
DOI: 10.1016/0095-8956(89)90063-4
author: Thomas Zaslavsky
published in: Journal of Combinatorial Theory. Series B
publication date: 1989-01-01T00:00:00Z
review text: A biased graph is a graph with a class of its circles (simple closed paths), called balanced, having the property that, whenever the union of two balanced circles is a theta graph, the third circle in the union is also balanced. A gain graph is a graph in which each edge has a gain, i.e. a value from a group so that reversing the direction inverts the gain, a circle is balanced if its edge gain product is the identity of the group. So, a gain graph defines a biased graph. (The gain graphs form a proper subclass of biased graphs.) This paper is the first part of the series in which the general structural and enumerative theory of biased-graphic matroids is developed. In this part, the fundamentals are presented of balance and minors (subgraphs and contractions) for biased and gain graphs, the parts that are presupposed by later parts. There is also the catalogue of the most interesting general types of biased and gain graphs followed by examples.
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 4174986
zbMATH Keywords: contraction
zbMATH Keywords: minor
zbMATH Keywords: biased graph
zbMATH Keywords: circles
zbMATH Keywords: paths
zbMATH Keywords: gain graph
zbMATH Keywords: biased- graphic matroids
reviewed by: Marie Demlová
MaRDI profile type: Publication
full work available at URL: 0095-8956(89)90063-4
OpenAlex ID: W2063835992
cites work: Q5331549
cites work: Q5619114
cites work: An interrelation between line graphs, eigenvalues, and matroids
cites work: A class of geometric lattices based on finite groups
cites work: On the notion of balance of a signed graph
cites work: On local balance and \(n\)-balance in signed graphs
cites work: Matroids from directed graphs
cites work: Q3863425
cites work: On subgraphs as matroid cells
cites work: On matroids on edge sets of graphs with connected subgraphs as circuits. II
cites work: Lectures on matroids
cites work: The Geometry of Root Systems and Signed Graphs
cites work: Characterizations of signed graphs
cites work: Signed graphs
cites work: Signed graph coloring
cites work: Chromatic invariants of signed graphs
cites work: Biased graphs. II: The three matroids
cites work: Biased graphs. III: Chromatic and dichromatic invariants
cites work: The biased graphs whose matroids are binary
description: scientific article; zbMATH DE number 4174986