Related papers: The Cayley graph of a quandle

It is known that the automorphism group of the elementary abelian $2$-group $Z_2^n$ is isomorphic to the general linear group $GL(n,F_2)$ of degree $n$ over $F_2$. Let $W$ be the collection of permutation matrices of order $n$. It is clear…

A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of…

We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group -- the part of this bipartite Cayley…

We introduce and investigate a natural family of metrics on connected components of a rack. The metrics are closely related to certain bi-invariant metrics on the group of inner automorphisms of the rack. We also introduce a bounded…

A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been…

We consider the family of undirected Cayley graphs associated with odd cyclic groups, and study statistics for the eigenvalues in their spectra. Our results are motivated by analogies between arithmetic geometry and graph theory.

We study when the adjacency matrix of a Cayley graph factors as the product of two adjacency matrices of Cayley graphs. Let $G$ be a finite group and let $U\subseteq G\setminus \{e\}$ be symmetric. Writing $A(G;U)$ for the adjacency matrix…

From a generalization to $Z^n$ of the concept of congruence we define a family of regular digraphs or graphs called multidimensional circulants, which turn out to be Cayley (di)graphs of Abelian groups. This paper is mainly devoted to show…

Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at least $\eta d$. This says that if we choose a random vertex $x\in A$ and a random…

The isomorphism problem for digraphs is a fundamental problem in graph theory. This problem for Cayley digraphs has been extensively investigated over the last half a century. In this paper, we consider this problem for $m$-Cayley digraphs…

A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are…

In this paper, we investigate a quandle structure induced by an augmented rack arising from a gauge transformation group. We construct a quandle from a principal bundle and its discrete generalization. When we see a group as a (discrete)…

In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differential calculi (over the algebra of functions) on a discrete…

We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost…

Let $G$ be a group. The directed endomorphism graph, $\dend(G)$ of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex $a$ to the vertex $b$ if $a \neq b$ and there exists an endomorphism on $G$ mapping…

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 study rack and quandle coverings from a universal algebraic viewpoint and we show how they can be understood using the notion of strongly abelian congruences. We provide an abstract characterization of several particular types of…

Quandle representations are homomorphisms from a quandle to the group of invertible matrices on some vector space taken with the conjugation operation. We study certain families of quandle representations. More specifically, we introduce…

It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. Here we prove that a discrete version of this property (called local to global rigidity) holds for a…

This paper deals with the Cayley graph $\Cay,$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. We prove that…