Related papers: Automorphisms of infinite Johnson graph

The Johnson graph $J(n,i)$ is defined to the graph whose vertex set is the set of all $i$-element subsets of $\{1,\ldots,n\}$, and two vertices are joined whenever the cardinality of their intersection is equal to $i-1$. In Ramras and…

The Johnson graph $J(n, i)$ is defined as the graph whose vertex set is the set of all $i$-element subsets of $\{1, . . ., n \}$, and two vertices are adjacent whenever the cardinality of their intersection is equal to $i$-1. In Ramras and…

Extending earlier results of Godsil and of Dobson and Malnic on Johnson graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are Cayley graphs, that is, which are connected and have a group of automorphisms acting…

The Johnson graph J(n,N) is defined as the graph whose vertices are the n-subsets of the set {1,2,...,N}, where two vertices are adjacent if they share exactly n - 1 elements. Unlike Johnson graphs, induced subgraphs of Johnson graphs (JIS…

The partial automorphisms of a graph $X$ having $N$ vertices are the bijections $\sigma:I\to J$ with $I,J\subset\{1,\ldots,N\}$ which leave invariant the edges. These bijections form a semigroup $\widetilde{G}(X)$, which contains the…

An automorphism of a graph $G=(V,E)$ is a bijective map $\phi$ from $V$ to itself such that $\phi(v_i)\phi(v_j)\in E$ $\Leftrightarrow$ $v_i v_j\in E$ for any two vertices $v_i$ and $v_j$. Denote by $\mathfrak{G}$ the group consisting of…

When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More…

Suppose $X$ is a simple graph. The $X-$join $\Gamma$ of a set of complete or empty graphs $\{X_x \}_{x \in V(X)}$ is a simple graph with the following vertex and edge sets: \begin{eqnarray*} V(\Gamma) &=& \{(x,y) \ | \ x \in V(X) \ \& \ y…

Let $G=(V,E)$ be a graph with the vertex-set $V$ and the edge-set $E$. Let $N(v)$ denote the set of neighbors of the vertex $v$ of $G.$ The graph $G$ is called $ irreducible $ whenever for every $v,w \in V$ if $v \neq w$, then $N(v)\neq…

An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively…

In this article we generalize a theorem of Benson for generalized quadrangles to strongly regular graphs and directed strongly regular graphs. The main result provides numerical restrictions on the number of fixed vertices and the number of…

An automorphism of a graph $G$ with $n$ vertices is a bijective map $\phi$ from $V(G)$ to itself such that $\phi(v_i)\phi(v_j)\in E(G)$ $\Leftrightarrow$ $v_i v_j\in E(G)$ for any two vertices $v_i$ and $v_j$ of $G$. Denote by…

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular…

We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant…

In a recent paper [Comm. Algebra, 44(2016) 4724-4731], Das introduced the graph $\mathcal{I}n(\mathbb{V})$, called subspace inclusion graph on a finite dimensional vector space $\mathbb{V}$, where the vertex set is the collection of…

For positive integers $n,k$ and $t$, the uniform subset graph $G(n, k, t)$ has all $k$-subsets of $\{1,2,\ldots, n\}$ as vertices and two $k$-subsets are joined by an edge if they intersect at exactly $t$ elements. The Johnson graph…

Using the canonical JSJ splitting, we describe the outer automorphism group $\Out(G)$ of a one-ended word hyperbolic group $G$. In particular, we discuss to what extent $\Out(G)$ is virtually a direct product of mapping class groups and a…

We characterise connected cubic graphs admitting a vertex- transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a…

We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or…