Related papers: On the Automorphism Group of a Graph

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of…

Let $\psi : G\to GL(V)$ and $\varphi :G \to GL (W)$ be representations of finite group $G$. A linear map $T: V\to W$ is called a morphism from $\psi$ to $\varphi$ if it satisfys $T\psi_g= \varphi_g T$ for each $g\in G$ and let…

Let $G$ be a group. \textit{The permutability graph of cyclic subgroups of $G$}, denoted by $\Gamma_c(G)$, is a graph with all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\Gamma_c(G)$ are adjacent if and…

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…

A quantum graph $\mathcal{G}$ housed by a matrix algebra $M_n$ can be encoded as an operator system $\mathcal S=\mathcal{S}_{\mathcal{G}}\le M_n$. There are two sensible notions of quantum automorphism group for any such:…

Given an affine algebraic variety $X$, we prove that if the neutral component $\mathrm{Aut}^\circ(X)$ of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves…

We describe the outer automorphism group of a one-ended fundamental group of a graph of groups, when edge groups are cyclic, and vertex groups are torsion-free with cyclic centralizers. We show that in this case the outer automorphism group…

This article initiates a geometric study of the automorphism groups of general graph products of groups, and investigates the algebraic and geometric structure of automorphism groups of cyclic product of groups. For a cyclic product of at…

Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be…

We define several "standard" subgroups of the automorphism group Aut(G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut(G). If C is the commutation graph of G, we show…

Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group…

It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…

Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops is obtained as follows: Let $m$ be a positive even integer, $G$ an abelian group, and $\alpha$ an automorphism of $G$ that…

For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic,…

A graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let $\Gamma$ be a connected graph…

The aim of this paper is to use the framework of incidence geometry to develop a theory that permits to model both the inner and outer automorphisms of a group G simultaneously. More precisely, to any group G, we attempt to associate an…

Let $G$ be a finite group and $\sigma$ a partition of the set of all? primes $\Bbb{P}$, that is, $\sigma =\{\sigma_i \mid i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_i$ and $\sigma_i\cap \sigma_j= \emptyset $ for all $i\ne j$. If $n$…

A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where…

The $G$-graph $\Gamma(G,S)$ is a graph from the group $G$ generated by $S\subseteq G$, where the vertices are the right cosets of the cyclic subgroups $\langle s \rangle, s\in S$ with $k$-edges between two distinct cosets if there is an…

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…