Research project is (co) funded by the Slovenian Research Agency.

**UL Member:** Faculty of Mathematics and Physics

**Code:** N1-0216

**Project:** Symmetry, fixity and flexibility of graphs

**Period:** 1. 10. 2021 - 30. 9. 2024

**Range per year:** 0,8 FTE **category:** B

**Head:** Primož Potočnik

**Research activity**: Natural sciences and mathematics

**Research Organisations, researchers, citations for bibliographic records**

**Project description:**

This project is about symmetry. Broadly speaking, we propose to consider the following set of questions: How symmetrical can a mathematical object be? What influence does a high level of symmetry have on the properties of the object? Which structural properties of an object prevent it to from being highly symmetrical and which guarantee a high level of symmetry?

In mathematics, an object is usually viewed as highly symmetrical provided that it has a ``rich'' automorphism group. The notion of "rich" may vary from "having large order" to "acting in a specified way" (for example, transitively) on the constitutive parts of the object. The focus of the project, however, is on less studied concepts, which we call "flexibility" and "fixity".

For a discrete object (such as a graph), the fixity of the object is defined as the maximum number of the underlying points fixed by a nontrivial symmetry. An object is then considered flexible provided that it has "large" fixity (where the meaning of "large" varies in different concrete contexts). Studying fixity has a long history in the setting of permutation groups but surprisingly, not much is known about it in the setting of symmetry groups acting on discrete objects.

The goal of this research project is to initiate a systematic and in-depth investigation of this phenomenon with the emphasis on the relationship between the fixity and the structural restrictions on the objects. Besides curiosity and a fundamental desire to understand the nature of this concept, a major driving force for the proposed research comes from our desire to make significant progress towards some long-standing conjectures and problems in the area of graph theory, such as the polycirculant conjecture. We believe that the concept of fixity can be used in tackling with some of these problems.