### Basic Research Project J1-2451

Symmetries in Graphs via Rigid Cells

Project Title: Symmetries in Graphs via Rigid Cells

PI: Dragan Marušič.

Project Code: J1-2451

Type of the Project: basic research project.

Funding Organization: Slovenian Research Agency (ARRS).

Research Field (ARRS): 1.01.00 - Natural Sciences and Mathematics / Mathematics.

Duration: 1. 9. 2020 - 31. 8. 2023.

Project Category: B.

Yearly Range: 1.60 FTE (2.726 research hours).

Sicris profile of the project is avaliable __here__.

Research Organizations:

University of Primorska, Andrej Marušič Institute,

University of Primorska, Faculty of Mathematics, Natural Sciences and Information Technologies,

University of Ljubljana, Faculty of Education.

Project Members:

Marušič Dragan (leader) (ARRS code: 02887)

Dobson Edward Tauscher (ARRS code: 34109)

Fernandez Blas (ARRS code: 52892)

Hujdurović Ademir (ARRS code: 32518)

Kovacs Istvan (ARRS code: 25997)

Kudin Sadmir (ARRS code: 51980)

Kutnar Klavdija (ARRS code: 24997)

Malnič Aleksander (ARRS code: 02507)

Miklavič Štefko (ARRS code: 21656)

Morgan Luke (ARRS code: 52908)

Pasalic Enes (ARRS code: 27777)

Požar Rok (ARRS code: 32026)

Rodriguez Rene (ARRS code: 52701)

Šparl Primož (ARRS code: 23341)

Velkavrh Žiga (ARRS code: 50720)

Woodroofe Russell Stephen (ARRS code: 50355)

Abstract:

When dealing with symmetries in graphs different features of their automorphism groups have been studied over the years. Let us take, for example, the well-known and still open polycirculant conjecture which states that every vertex-transitive (di)graph admits a derangement of prime order, that is, an automorphism of prime order without fixed vertices. The results obtained thus far suggest that certain important properties of vertex-transitive graphs are reflected in and may be deduced from such automorphisms. At the other extreme one might want to study those automorphisms which do fix at least one vertex of a vertex-transitive graph, that is, automorphisms which belong to vertex stabilizers. In this context an immediate natural question arises:

**What other additional vertices will such an automorphism fix? More precisely, what is the structure of the subgraph induced by all the vertices fixed by this automorphsim? **

This question is the essential ingredient of the proposed project. The subgraphs induced by the set of all fixed vertices of a given automorphism will be referred to as rigid subgraphs, and a connected component of such a subgraph will be called a rigid cell. We will adopt a mixed strategy approach to the symmetry problem by combining group-theoretic and graph-theoretic tools. New insights into inner structure of vertex-transitive graphs and other classes of graphs satisfying certain specific symmetry conditions, are expected. The following main lines of research will be pursued within this project proposal:

- Studying the structure of rigid cells in vertex-transitive graphs.
- Studying the structure of automorphisms giving rise to rigid cells. In particular, addressing the following question: under what conditions automorphisms of the same order belong to the same conjugacy class in the automorphism group?
- Finding combinatorial (graph-theoretic) reflections of the concept of real (strongly real) group elements (in particular with regards to consistent cycles in graphs) , where an element of a group is real if it belongs to the same conjugacy class as its inverse, and is strongly real if it is conjugate to its inverse via an involution.

Finally, in line with certain opinions in mathematical community that a more conservative use of the Classification of finite simple groups (CFSG) in various problems in algebraic graph theory and the theory of permutation groups should be adopted, attempts will be made to find direct proofs of certain theorems in whose completion the CFSG had played an essential role.