Projects / Programmes source: ARIS

Symmetries in graphs via rigid cells

Research activity

Code Science Field Subfield
1.01.05  Natural sciences and mathematics  Mathematics  Graph theory 

Code Science Field
1.01  Natural Sciences  Mathematics 
vertex-transitive graph, arc-transitive graph, rigid cell, symmetry, consistent cycle, (strongly) real group element
Evaluation (rules)
source: COBISS
Researchers (18)
no. Code Name and surname Research area Role Period No. of publicationsNo. of publications
1.  34109  PhD Edward Tauscher Dobson  Mathematics  Researcher  2020 - 2023  74 
2.  52892  PhD Blas Fernandez  Mathematics  Junior researcher  2021 - 2022  22 
3.  32518  PhD Ademir Hujdurović  Mathematics  Researcher  2020 - 2023  102 
4.  25997  PhD Istvan Kovacs  Mathematics  Researcher  2020 - 2023  213 
5.  51980  PhD Sadmir Kudin  Mathematics  Junior researcher  2020 - 2022 
6.  24997  PhD Klavdija Kutnar  Mathematics  Researcher  2020 - 2023  249 
7.  23501  PhD Boštjan Kuzman  Mathematics  Researcher  2021 - 2023  249 
8.  02507  PhD Aleksander Malnič  Mathematics  Researcher  2020 - 2023  247 
9.  02887  PhD Dragan Marušič  Mathematics  Head  2020 - 2023  597 
10.  21656  PhD Štefko Miklavič  Mathematics  Researcher  2020 - 2023  201 
11.  52908  PhD Graham Luke Morgan  Mathematics  Researcher  2020 - 2022  38 
12.  27777  PhD Enes Pasalic  Mathematics  Researcher  2020 - 2023  131 
13.  32026  PhD Rok Požar  Mathematics  Researcher  2020 - 2023  42 
14.  52701  PhD Rene Rodriguez  Mathematics  Researcher  2020 - 2023 
15.  57037  Ksenija Rozman  Mathematics  Researcher  2022 - 2023 
16.  23341  PhD Primož Šparl  Mathematics  Researcher  2020 - 2023  185 
17.  50720  PhD Žiga Velkavrh  Mathematics  Junior researcher  2020 - 2021  14 
18.  50355  PhD Russell Stephen Woodroofe  Mathematics  Researcher  2020 - 2023  80 
Organisations (3)
no. Code Research organisation City Registration number No. of publicationsNo. of publications
1.  0588  University of Ljubljana, Faculty of Education  Ljubljana  1627082  30,242 
2.  1669  University of Primorska, Andrej Marušič Insitute  Koper  1810014007  10,463 
3.  2790  University of Primorska, Faculty of mathematics, Natural Sciences and Information Technologies  Koper  1810014009  17,262 
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.
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