# Graphs, groups, and more: celebrating Brian Alspach’s 80th and Dragan Marušič’s 65th birthdays

from 28 May 2018 to 1 June 2018
Koper
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# Some problems about symmetries of finite graphs

## Speakers

• Mr. Primož POTOČNIK

## Content

A few topics regarding symmetries of finite graphs that I find interesting, intriguing and worth studying shall be presented.

## Summary

I would like to mention a few topics regarding symmetries of finite graphs that I find interesting, intriguing and worth studying.

The first topic is about lifting automorphisms along covering projections. Suppose one is given a finite connected graph $\Gamma$ and a group of automorphisms $G$ acting on it. Can one find a regular covering projection $\wp$ onto $\Gamma$ such that $G$ is the maximal group that lifts along $\wp$ and such that the full automorphism group of the graph is the lift of $G$? A recent partial result answer proved recently my Pablo Spiga and myself will be presented.

The second topic is about vertex-transitive graphs admitting an automorphism fixing many vertices; here a strict definition of the term many'' is intentionally avoided so that by varying it one can prove different results. Some computational data regarding cubic vertex-transitive graphs will be presented.

If time permits, a third topic regarding vertex-transitive graphs admitting an automorphism with a long orbit will be discussed; here the term long'' means a suitable fixed proportion of the order of the graph. Some results obtained recently by Micael Toledo about the cubic case will be mentioned.