Finite 33-connected homogeneous graphs

A finite graph \G is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is $2$--transitive or G_v^{\G(v)} is of rank $3$ and \G has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types

The vertex-transitive TLF-planar graphs

We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs, a class containing in particular all the one-ended planar Cayley graphs and the normal transitive tilings. We characterize these graphs with a finite local representation and a special kind of finite state automaton named labeling scheme. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of any given degree. Also, we are able decide to whether any TLF-planar transitive graph is Cayley or not.Comment: Article : 23 pages, 15 figures Appendix : 13 pages, 72 figures Submitted to Discrete Mathematics The appendix is accessible at http://www.labri.fr/~renault/research/research.htm

Monotonicity for continuous-time random walks

Consider continuous-time random walks on Cayley graphs where the rate assigned to each edge depends only on the corresponding generator. We show that the limiting speed is monotone increasing in the rates for infinite Cayley graphs that arise from Coxeter systems, but not for all Cayley graphs. On finite Cayley graphs, we show that the distance -- in various senses -- to stationarity is monotone decreasing in the rates for Coxeter systems and for abelian groups, but not for all Cayley graphs. We also find several examples of surprising behaviour in the dependence of the distance to stationarity on the rates. This includes a counterexample to a conjecture on entropy of Benjamini, Lyons, and Schramm. We also show that the expected distance at any fixed time for random walks on $\mathbb{Z}^+$ is monotone increasing in the rates for arbitrary rate functions, which is not true on all of $\mathbb{Z}$. Various intermediate results are also of interest.Comment: 29 pp., 1 fi

Rotation groups, mediangle graphs, and periagroups: a unified point of view on Coxeter groups and graph products of groups

In this article, we introduce rotation groups as a common generalisation of Coxeter groups and graph products of groups (including right-angled Artin groups). We characterise algebraically these groups by presentations (periagroups) and we propose a combinatorial geometry (mediangle graphs) to study them. As an application, we give natural and unified proofs for several results that hold for both Coxeter groups and graph products of groups.Comment: 39 pages, comments are welcome

Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite 22-arc-transitive graph which is not a Cayley graph

A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper we give a sufficient condition such that the automorphism group of the Cayley graph \Cay(H,(X\cup Y)\setminus\{1\}) is equal to $H: A(H,X,Y)$, where $A(H,X,Y)$ is the setwise stabiliser in \Aut(H) of $X\cup Y$. We use this criterion to resolve a questions of Li, Ma and Pan from 2009, by constructing a $2$-arc transitive normal cover of order $2^{53}$ of the complete bipartite graph \K_{16,16} and prove that it is \emph{not} a Cayley graph.Comment: arXiv admin note: text overlap with arXiv:2303.00305, arXiv:2211.1680