Faculty of Mathematics, Physics
and Informatics
Comenius University Bratislava

Algebraic Graph Theory Seminar - Štefan Gyürki (6.10.2023)

Friday 6.10.2022 at 13:00, Lecture room M/VIII (online too)

03. 10. 2023 12.01 hod.
By: Martin Mačaj

Štefan Gyürki (Slovak University of Technology, Bratislava):
Automorphisms of a family of small $(q,8)$-graphs

For given integers $k\geq 2$ and $g\geq 3$ the $k$-regular graphs of girth $g$ are called $(k,g)$-graphs. In the cage problem one has to construct the smallest possible $(k,g)$-graph (with respect to the order). The smallest such graphs are called $(k,g)$-cages. It is known that the $(q+1,8)$-cages, when $q$ is an odd prime power, are arising as incidence graphs of generalized quadrangles, thus they are very symmetric in the sense of automorphisms and transitivity. There were a few attempts to construct small $(q,8)$-graphs from the $(q+1,8)$-cages as induced subgraphs. In this talk, maybe surprisingly, we show that a family of such $(q,8)$-graphs of order $2q(q^2-1)$ is not so symmetric in comparison with other families. More precisely, we show that their group of automorphisms has precisely $4$ orbits on the set of vertices.

Joint work with Pavol Jánoš

Those of you who are not able to attend in person or who are still uncertain about the safety of attending in person are welcome to attend via MS Teams. In either case, we hope to see as many of you as possible (either in person or virtually) at our Friday gatherings.