Related papers: A study on Type-2 isomorphic circulant graphs and …

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of $\mathrm{PSL}_{n}(q)$…

We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F_n) \to Out(F_m). We prove that if n > 8 is even and n \neq m \leq 2n, or n is odd and n \neq m \leq 2n - 2, then all such homomorphisms have…

Let $G = H\times A$ be a group, where $H$ is a purely non-abelian subgroup of $G$ and $A$ is a non-trivial abelian factor of $G$. Then, for $n \geq 2$, we show that there exists an isomorphism $\phi : Aut_{Z(G)}^{\gamma_{n}(G)}(G)…

A divisor graph $G$ is an ordered pair $(V, E)$ where $V \subset \mathbbm{Z}$ and for all $u \neq v \in V$, $u v \in E$ if and only if $u \mid v$ or $v \mid u$. A graph which is isomorphic to a divisor graph is also called a divisor graph.…

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…

The number of maximal independent sets of the n-cycle graph C_n is known to be the nth term of the Perrin sequence. The action of the automorphism group of C_n on the family of these maximal independent sets partitions this family into…

We consider isomorphisms and automorphisms of quantum groups. Let $k$ be a field and suppose $p, q\in k^*$ are not roots of unity. We prove that the two quantum groups $U_q(\mathfrak {sl}_2)$ and $U_p(\mathfrak{sl}_2)$ over a field $k$ are…

A loop is said to be automorphic if its inner mappings are automorphisms. For a prime $p$, denote by $\mathcal A_p$ the class of all $2$-generated commutative automorphic loops $Q$ possessing a central subloop $Z\cong \mathbb Z_p$ such that…

Let $G$ be a nonabelian group, $A\subseteq G$ an abelian subgroup and $n\geqslant 2$ an integer. We say that $G$ has an $n$-abelian partition with respect to $A$, if there exists a partition of $G$ into $A$ and $n$ disjoint commuting…

We prove that $Aut({\mathbb S}^1)$ coincides with the automorphism group of the \emph{circle graph} $\mathcal{C}$, i.e. the intersection graph of the family of chords of ${\mathbb S}^1$. We prove that the countable subgraph of $\mathcal{C}$…

We construct explicity the automorphism group of the folded hypercube $FQ_n$ of dimension $n>3$, as a semidirect product of $N$ by $M$, where $N$ is isomorphic to the Abelian group $Z_2^n$, and $M$ is isomorphic to $Sym(n+1)$, the symmetric…

This paper considers non-Abelian homology groups of a group diagram introduced as homotopy groups of a simplicial change. We prove a theorem stating that the non-Abelian homology groups of a group diagram are isomorphic to the homotopy…

The canonical double cover $B(X)$ of a graph $X$ is the direct product of $X$ and $K_2$. If $Aut(B(X)) \cong Aut(X) \times \mathbb{Z}_2$ then $X$ is called stable; otherwise $X$ is called unstable. An unstable graph is nontrivially unstable…

Let $G$ be a finite group. For each $m>1$ we define the symmetric canonical subset $S=S(m)$ of the Cartesian power $G^m$ and we consider the family of Cayley graphs $\mathscr{G}_m(G)=Cay(G^m,S)$. We describe properties of these graphs and…

The algebraic degree $Deg(G)$ of a graph $G$ is the dimension of the splitting field of the adjacency polynomial of $G$ over the field $\mathbb{Q}$. It can be shown that for every positive integer $d$, there exists a circulant graph with…

Presentations for the holomorphs of abelian groups of the form $C_{p^n} \times 1^{m}$ for $p$=2 or an odd prime are given. These presentations extend the results given in Burnside's well-known text on finite groups on the holomorphs for the…

A graph is pseudo 2-factor isomorphic if all of its 2-factors have the same parity of number of cycles. Abreu et al. [J. Comb. Theory, Ser. B. 98 (2008) 432--442] conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the…

Let D be the circulant digraph with n vertices and connection set {2,3,c}. (Assume D is loopless and has outdegree 3.) Work of S.C.Locke and D.Witte implies that if n is a multiple of 6, c is either (n/2) + 2 or (n/2) + 3, and c is even,…

Let $\Gamma=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(\Gamma)$. The Cayley index of $\Gamma$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a…

Combining the the results of A.R. Meyer and L.J. Stockmeyer "The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space", and K.S. Booth "Isomorphism testing for graphs, semigroups, and finite automata are…