# Klavdija Kutnar: Vertex-transitive graphs: from derangements to rigid cells

Dear colleagues,

The organizers of the Algebraic Graph Theory International Webinar would like to invite you to join us and other colleagues on February 2, 2021, at 7pm Central European Time, for the next presentation delivered by **Klavdija Kutnar**.

She will speak on **Vertex-transitive graphs: from derangements to rigid cells**

**Abstract:** When dealing with symmetry of combinatorial objects -- or in any other setting for that matter -- one will inexorably come across two different kinds of nonidentity automorphisms of these objects: those fixing as large as possible subset of points, on the one hand, and those fixing no points at all on the other.
For example, given a graph X and an automorphism a of X, we let Fix(a) be the set of all those vertices of X which are fixed by a. When Fix(a) is empty, then a is called a derangement, and furthermore, it is said to be semiregular if all of its cycles in the cycle decomposition are of the same length. When Fix(a) is non-empty, then an a-rigid cell is a connected component of the subgraph induced by Fix(a). In this lecture, I will aim for two goals:

G1: give a complete description of rigid cells in cubic arc-transitive graphs, and

G2: discuss some recent developments in regards to the problem of deciding which cubic arc-transitive graphs admit the so called simplicial automorphisms, that is, semiregular auto-morphisms whose quotient ``multigraphs'' are in fact simple graphs -- a special case of the well known semiregularity problem regarding existence of semiregular automorphisms in vertex-transitive graphs.

Further details may be found at http://euler.doa.fmph.uniba.sk/AGTIW.html where you can also find the slides and the recordings of our previous presentations. Also, if you wish to advertise an AGT friendly conference on this page, please send us the link.

Hoping to see you at the webinar, and wishing you all the best. Stay AGTIW.

Isabel Hubard, Robert Jajcay and Primoz Potočnik