Related papers: Quantum automorphism groups of finite graphs

Let $k \geq 1$ be an integer and $n=3k-1$. Let $\mathbb{Z}_n$ denote the additive group of integers modulo $n$ and let $C$ be the subset of $\mathbb{Z}_n$ consisting of the elements congruent to 1 modulo 3. The Cayley graph…

Gauge theories are important descriptions for many physical phenomena and systems in quantum computation. Automorphism of gauge group naturally gives global symmetries of gauge theories. In this work we study such symmetries in gauge…

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this…

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all…

Every indefinite binary form occurs as the Picard lattice of some K3-surface. The group of its isometries, or automorphs, coincides with the automorphism group of the K3-surface, but only up to finite groups. The classical theory of…

For a finite group $G$, denote by $\alpha(G)$ the minimum number of vertices of any graph $\Gamma$ having $\text{Aut}(\Gamma)\cong G$. In this paper, we prove that $\alpha(G)\leq |G|$, with specified exceptions. The exceptions include four…

This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if $\Gamma$ is a finite graph which contains at least two vertices and is not a join and…

A circulant (di)graph is a (di)graph on n vertices that admits a cyclic automorphism of order n. This paper provides a survey of the work that has been done on finding the automorphism groups of circulant (di)graphs, including the…

We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical…

The present paper constructs unbounded quasimorphisms that are invariant under all automorphisms on free products of more than two factors and on graph products of finitely generated abelian groups. This includes many classes of right…

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.

The quantum mechanics is proved to admit no hidden-variable in 1960s, which means the quantum systems are contextual. Revealing the mathematical structure of quantum mechanics is a significant task. We develop the approach of partial…

We determine the automorphism group of the zero-divisor digraph of the semiring of matrices over an antinegative commutative semiring with a finite number of zero-divisors.

Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the…

We describe computational results about undirected graphs having $10$ vertices and automorphism group isomorphic to $\mathbb{Z}/4\mathbb{Z}$.

A free wreath product construction of a Hopf algebra (or of a Woronowicz algebra) by Wang's quantum permutation group is done. It provides new examples of quantum groups and is useful to describe the quantum automorphism group of the…

In this paper, we have computed the automorphism groups of all groups of order $p^{2}q^{2}$, where $p$ and $q$ are distinct primes.

We work out axioms for the duals $G\subset U_N^+$ of the finite quantum permutation groups, $F\subset S_N^+$ with $|F|<\infty$, and we discuss how the basic theory of such quantum permutation groups partly simplifies in the dual setting. We…

A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the…

Let $G$ be a finite group and let $S$ be an inverse-closed subset of $G$ not containing the identity. The Cayley graph $\mathrm{Cay}(G,S)$ has vertex set $G$, where two vertices $x$ and $y$ are adjacent if and only if $x^{-1}y \in S$.…