# CSASC 2013

9-13 June 2013
Koper, Slovenia
UTC timezone
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# CAYLEY GRAPHS ON ABELIAN GROUPS

Presented by Mr. Gabriel VERRET
Type: Oral presentation
Track: Symmetries in Graphs, Maps and Other Discrete Structures

## Content

Let $G$ be a group and let $S$ be an inverse-closed subset of $G$. The \emph{Cayley graph} $\mathrm{Cay}(G,S)$ is the graph with vertex-set $G$ and with $g$ being adjacent to $h$ if and only if $gh^{-1}\in S$. It is easy to see that $G$ acts regularly as a group of automorphisms of $\mathrm{Cay}(G,S)$ by right multiplication. Moreover, if $G$ is abelian and $\iota$ is the automorphism of $G$ mapping every element to its inverse, then $\iota$ acts an as automorphism of $\mathrm{Cay}(G,S)$ fixing the identity. We prove that, in an appropriate sense, almost all Cayley graphs on an abelian group $G$ have $G\rtimes\langle\iota\rangle$ as their full automorphism group. This settles a conjecture of Babai and Godsil from 1982. This is joint work with Ted Dobson and Pablo Spiga.

## Place

Location: Koper, Slovenia

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