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TY  - THES
AU  - Limbach, Anna Margarethe
TI  - On the clique dynamics of locally cyclic graphs
PB  - RWTH Aachen University
VL  - Dissertation
CY  - Aachen
M1  - RWTH-2024-05608
SP  - 1 Online-Ressource : Illustrationen
PY  - 2024
N1  - Veröffentlicht auf dem Publikationsserver der RWTH Aachen University
N1  - Dissertation, RWTH Aachen University, 2024
AB  - In this thesis, it is proven that the clique graph operator k is divergent on a (notnecessarily finite) locally cyclic graph G with minimum degree δ ≥ 6 if and onlyif the universal triangular cover of G contains arbitrarily large triangular-shaped subgraphs. For finite G, this is equivalent to G being 6-regular. A graph is called locally cyclic if the open neighbourhood N G (v) of each vertex v induces a cycle. The clique graph kG of a graph G has the maximal complete subgraphs of G as its vertices and its edges are those pairs with non-empty intersection. The (n + 1)-st iterated clique graph is recursively defined as the clique graph of the n-th iterated clique graph. If all iterated clique graphs of Gare non-isomorphic, the graph G is called clique divergent; otherwise, it is clique convergent. While it has been shown for finite locally cyclic graphs that those with minimum degree δ ≥ 7 are clique convergent while the 6-regular ones are clique divergent, this thesis gives a full characterisation of clique convergent locally cyclic graphs with minimum degree δ ≥ 6.In the beginning, it is shown that a clique convergent connected graph has a clique convergent universal triangular cover. Conversely, a sufficient condition is given under which the clique convergence of the universal triangular cover of a graph implies the clique convergence of the graph itself. Locally cyclic graphs with minimum degree δ ≥ 6 which are triangularly simply connected are their own universal covers and they are referred to as pikas throughout this thesis. On the class of pikas, clique convergence is characterise dusing an explicit construction of the iterated clique graphs and a finite yet divergent parameter for the clique divergent case. Furthermore, it is shown thatlocally cyclic graphs with minimum degree δ ≥ 6 are clique convergent if and only if their universal covers are clique convergent. This way, the characterisation is completed.
LB  - PUB:(DE-HGF)11
DO  - DOI:10.18154/RWTH-2024-05608
UR  - https://publications.rwth-aachen.de/record/987344
ER  -