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Double Covers, Cancellation and Automorphisms.

Ponente: Wilfried Imrich
Institución: Department Mathematics and Information Technology, Montanuniversität Leoben
Tipo de Evento: Investigación, Divulgación
Cuándo 10/09/2018
de 12:00 a 13:00
Dónde Sala A2 del Centro Académico Cultural (CAC)
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The motivation for  this talk are the similarities of results  on automorphisms of double  covers  with those on the cancellation of graphs with respect to the direct product.  The \emph{double cover} of a graph $G$  is the direct product of $G$ by $K_2$, and $G$ is \emph{unstable} if the automorphism group of its double cover is larger than  $\operatorname{Aut}(G)\times \mathbb{Z}_2$. $G$ is called \emph{nontrivially unstable}, if it is non-bipartite and if any two distinct vertices have different neighborhoods, in other words, if $G$ is non-bipartite and thin.  1989 Maru\v{s}i\v{c}, Scapellato and Zagaglia Salvi proved  that to any nontrivially unstable graph $G$ there always exists a permutation matrix $P$ such that the product $AP$, where $A$ is the adjacency matrix of $G$, is the adjacency matrix of some graph. They asked when the converse holds.
Translated into the language of matrices  the answer of Mizzi and Scapellato form 2013 is that  $G$ is nontrivially unstable  if and only if there exists  a permutation matrix $P$ of order $>2$ such that $PAP=A$. We give a proof  using a result of Richard Hammack about antiautomorphisms and cancellation properties of direct products.
The talk also presents the background and discusses related results and problems about direct products.