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Group-theoretic generalisations of vertex and edge connectivities


Authors: Yinan Li and Youming Qiao
Journal: Proc. Amer. Math. Soc. 148 (2020), 4679-4693
MSC (2010): Primary 20D15, 05C40, 15A69
DOI: https://doi.org/10.1090/proc/15184
Published electronically: August 14, 2020
MathSciNet review: 4143386
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Abstract:

Let $p$ be an odd prime. Let $P$ be a finite $p$-group of class $2$ and exponent $p$, whose commutator quotient $P/[P,P]$ is of order $p^n$. We define two parameters for $P$ related to central decompositions. The first parameter, $\kappa (P)$, is the smallest integer $s$ for the existence of a subgroup $S$ of $P$ satisfying (1) $S\cap [P,P]=[S,S]$, (2) $|S/[S,S]|=p^{n-s}$, and (3) $S$ is centrally decomposable. The second parameter, $\lambda (P)$, is the smallest integer $s$ for the existence of a central subgroup $N$ of order $p^s$, such that $P/N$ is centrally decomposable.

While defined in purely group-theoretic terms, these two parameters generalise, respectively, the vertex and edge connectivities of graphs: For a simple undirected graph $G$, through the classical procedures of Baer (Trans. Amer. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lovász (B. Braz. Math. Soc., 1989), there is a $p$-group $P_G$ of class $2$ and exponent $p$ that is naturally associated with $G$. Our main results show that the vertex connectivity $\kappa (G)$ equals $\kappa (P_G)$, and the edge connectivity $\lambda (G)$ equals $\lambda (P_G)$.

We also discuss the relation between $\kappa (P)$ and $\lambda (P)$ for a general $p$-group $P$ of class $2$ and exponent $p$, as well as the computational aspects of these parameters. In particular, our main results imply that the $p$-group central decomposition algorithm of Wilson (J. Algebra and J. of Group Theory, 2009) can be used to solve the graph connectivity problem.


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Additional Information

Yinan Li
Affiliation: Centrum Wiskunde and Informatica and QuSoft, Science Park 123, 1098XG Amsterdam, Netherlands
MR Author ID: 1240769
ORCID: 0000-0002-5456-1319
Email: Yinan.Li@cwi.nl

Youming Qiao
Affiliation: Center for Quantum Software and Information, University of Technology, Sydney, Ultimo NSW 2007, Australia
MR Author ID: 938330
ORCID: 0000-0003-4334-1449
Email: Youming.Qiao@uts.edu.au

Keywords: $p$-groups of class $2$, graph connectivity, matrix spaces, bilinear maps
Received by editor(s): October 30, 2019
Received by editor(s) in revised form: March 24, 2020
Published electronically: August 14, 2020
Additional Notes: The first author was partially supported by ERC Consolidator Grant 615307-QPROGRESS.
The second author was partially supported by the Australian Research Council DP200100950.
Communicated by: Martin Liebeck
Article copyright: © Copyright 2020 American Mathematical Society