Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Valency Problems of Saxl Graphs
73
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $G$ be a permutation group on a set $Omega$ and recall that a base for $G$ is a subset of $Omega$ such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of $G$, denoted $Sigma(G)$, with vertex set $Omega$ and two vertices adjacent if they form a base. If $G$ is transitive, then $Sigma(G)$ is vertex-transitive and it is natural to consider its valency (which we refer to as the valency of $G$). In this paper we present a general method for computing the valency of any finite transitive group and we use it to calculate the exact valency of every primitive group with stabiliser a Frobenius group with cyclic kernel. As an application, we calculate the valency of every almost simple primitive group with an alternating socle and soluble stabiliser and we use this to extend results of Burness and Giudici on almost simple primitive groups with prime-power or odd valency.
rate research
Read More
A graph $Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $Aut(Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $Aut(Ga)$ then $Ga$ is called a normal Cayley graph of $T$. Let $r$ be an odd prime. Fang et al. cite{FMW} proved that, with a finite number of exceptions for finite simple group $T$, every connected symmetric Cayley graph of $T$ of valency $r$ is normal. In this paper, employing maximal factorizations of finite almost simple groups, we work out a possible list of those exceptions for $T$.
Let $G$ be a transitive permutation group on a finite set $Omega$ and recall that a base for $G$ is a subset of $Omega$ with trivial pointwise stabiliser. The base size of $G$, denoted $b(G)$, is the minimal size of a base. If $b(G)=2$ then we can study the Saxl graph $Sigma(G)$ of $G$, which has vertex set $Omega$ and two vertices are adjacent if they form a base. This is a vertex-transitive graph, which is conjectured to be connected with diameter at most $2$ when $G$ is primitive. In this paper, we combine probabilistic and computational methods to prove a strong form of this conjecture for all almost simple primitive groups with soluble point stabilisers. In this setting, we also establish best possible lower bounds on the clique and independence numbers of $Sigma(G)$ and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of $Sigma(G)$.
A graph $Gamma$ is $k$-connected-homogeneous ($k$-CH) if $k$ is a positive integer and any isomorphism between connected induced subgraphs of order at most $k$ extends to an automorphism of $Gamma$, and connected-homogeneous (CH) if this property holds for all $k$. Locally finite, locally connected graphs often fail to be 4-CH because of a combinatorial obstruction called the unique $x$ property; we prove that this property holds for locally strongly regular graphs under various purely combinatorial assumptions. We then classify the locally finite, locally connected 4-CH graphs. We also classify the locally finite, locally disconnected 4-CH graphs containing 3-cycles and induced 4-cycles, and prove that, with the possible exception of locally disconnected graphs containing 3-cycles but no induced 4-cycles, every finite 7-CH graph is CH.
Let $G$ be a group and $Ssubseteq G$ its subset such that $S=S^{-1}$, where $S^{-1}={s^{-1}mid sin S}$. Then {it the Cayley graph ${rm Cay}(G,S)$} is an undirected graph $Gamma$ with the vertex set $V(Gamma)=G$ and the edge set $E(Gamma)={(g,gs)mid gin G, sin S}$. A graph $Gamma$ is said to be {it integral} if every eigenvalue of the adjacency matrix of $Gamma$ is integer. In the paper, we prove the following theorem: {it if a subset $S=S^{-1}$ of $G$ is normal and $sin SRightarrow s^kin S$ for every $kin mathbb{Z}$ such that $(k,|s|)=1$, then ${rm Cay}(G,S)$ is integral.} In particular, {it if $Ssubseteq G$ is a normal set of involutions, then ${rm Cay}(G,S)$ is integral.} We also use the theorem to prove that {it if $G=A_n$ and $S={(12i)^{pm1}mid i=3,dots,n}$, then ${rm Cay}(G,S)$ is integral.} Thus, we give positive solutions for both problems 19.50(a) and 19.50(b) in Kourovka Notebook.
We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2epsilon)^{1/2p}$-uniformly Lipschitz) with $p$ varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal $p$ for which $L^p$-fixed point properties hold and the conformal dimension of the boundary. In the Gromov density model, we prove that for every $p_0 in [2, infty)$ for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on $L^p$-spaces that are $(2-2epsilon)^{1/2p}$-uniformly Lipschitz, and this for every $pin [2,p_0]$. To accomplish these goals we find new bounds on the first eigenvalue of the p-Laplacian on random graphs, using methods adapted from Kahn and Szemeredis approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to $L^p$-spaces previous results for Kazhdans Property (T) established by Zuk and Ballmann-Swiatkowski.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
