NDLI: Conformal Decomposition of Integral Tensions and Potentials of Signed Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Chen, Beifang
Copyright Year	2017
Abstract	Let $\Gamma$ be a subgroup of an integral chain group on a set $E$. A nonzero chain $g$ of $\Gamma$ is said to be conformally decomposable if there exist nonzero chains $g_1,g_2$ of $\Gamma$ such that $g=g_1+g_2$ and $g_1(e)g_2(e)\geq 0$ for all $e\in E$. For a signed graph $\Sigma$ with edge set $E$, there are two subgroups $F(\Sigma,{\Bbb Z})$ and $T(\Sigma,{\Bbb Z})$ of the 1-chain group $C_1(\Sigma,{\Bbb Z})$, known as the flow lattice and tension lattice of $\Sigma$. The conformally indecomposable flows of $F(\Sigma,{\Bbb Z})$ are classified in [B. Chen and J. Wang, https://arXiv.org/abs/1112.0642, 2011; B. Chen, J. Wang, and T. Zaslavsky, Discrete Math., 340 (2017), pp. 1271--1786] as signed-graphic circuit flows and a class of characteristic vectors of certain directed Eulerian cycle-trees. In this paper we classify conformally indecomposable tensions of $T(\Sigma,{\Bbb Z})$ as characteristic vectors of signed-graphic directed bonds and a class of characteristic vectors of directed semi-bonds and directed hyper-bonds. The half-spin structures ($\pm\frac{1}{2}$-potential functions) of $\Sigma$ correspond to characteristic vectors of directed hyper-bonds. A byproduct is the classification of conformally indecomposable integral potentials.
Sponsorship	Research Grants Council, University Grants Committee
Starting Page	2457
Ending Page	2478
Page Count	22
File Format	PDF
ISSN	08954801
DOI	10.1137/140973037
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	31
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-11-02
Access Restriction	Subscribed
Subject Keyword	Graphs and abstract algebra orientation tension circuit cut Signed and weighted graphs conformal decomposition indecomposable tension Flows in graphs bond uni-cut semi-bond signed graph Directed graphs (digraphs), tournaments cycle-tree hyper-bond flow classification of indecomposable tensions
Content Type	Text
Resource Type	Article
Subject	Mathematics

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