Torsion formulas for signed graphs

Authors:
Beifang Chen;Jue Wang
Affiliations:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong;Industrial Training Centre of Shenzhen Polytechnic, Shenzhen 518055, PR China
Venue:
Discrete Applied Mathematics
Year:
2010

Citing 3
Cited 0

Orientation of signed graphs

European Journal of Combinatorics
Graph Theory With Applications

Graph Theory With Applications
The flow and tension spaces and lattices of signed graphs

European Journal of Combinatorics

Quantified Score

Hi-index	0.04

Visualization

Abstract

Following our recent exposition on the algebraic foundations of signed graphs, we introduce bond (circuit) basis matrices for the tension (flow) lattices of signed graphs, and compute the torsions of such matrices and Laplacians. We present closed formulas for the torsions of the incidence matrix, the Laplacian, bond basis matrices, and circuit basis matrices. These formulas show that the torsions of all such matrices are powers of 2, and so imply that the matroids of signed graphs are representable over any field of characteristic not 2. A notable feature of using torsion is that the Matrix-Tree formula for ordinary graphs and Zaslavsky's formula for unbalanced signed graphs are unified into one Matrix-Basis formula in terms of the torsion of its Laplacian matrix, rather than in terms of its determinant, which vanishes for an ordinary graph unless one row is deleted from the incidence matrix.