Related papers: On the classification of triply-transitive strongl…

In this paper we classify cubic vertex-transitive graphs of girth $7$, based on their signature. Such a graph is either a truncation of an arc-transitive dihedral scheme on a $7$-regular graph, the skeleton of a rotary map of type…

Let $\Gamma$ be a finite X-symmetric graph with a nontrivial X-invariant partition $\mathcal {B}$ on $V(\Gamma)$ such that $\Gamma_{\mathcal {B}}$ is a connected (X,2)-arc-transitive graph and $\Gamma$ is not a multicover of…

A graph $\Gamma$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding…

In 1992, Terwilliger introduced the notion of the \emph{Terwilliger algebra} in order to study association schemes. The Terwilliger algebra of an association scheme $\mathcal{A}$ is the subalgebra of the complex matrix algebra, generated by…

A vertex transitive graph $\Gamma$ is said to be $2$-distance transitive if for each vertex $u$, the group of automorphisms of $\Gamma$ fixing the vertex $u$ acts transitively on the set of vertices at distance $1$ and $2$ from $u$, while…

In this paper, we deal with a generalization $\Gamma(\Omega,q)$ of the bipartite graphs $D(k,q)$ proposed by Lazebnik and Ustimenko, where $\Omega$ is a set of binary sequences that are adopted to index the entries of the vertices. A few…

We study strongly graded groupoids, which are topological groupoids $\mathcal G$ equipped with a continuous, surjective functor $\kappa: \mathcal G \to \Gamma$, to a discrete group $\Gamma$, such that $\kappa^{-1}(\gamma)\kappa^{-1}(\delta)…

Let $\Gamma$ be a simple undirected graph on a finite vertex set and let $A$ be its adjacency matrix. Then $\Gamma$ is {\it singular} if $A$ is singular. The problem of characterising singular graphs is easy to state but very difficult to…

Given a permutation group $G$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph with connection set the set of all derangements of $G$. We prove that, when $G$ is transitive of degree at least $3$, $\Gamma_G$ contains a triangle.…

The canonical double cover $\mathrm{D}(\Gamma)$ of a graph $\Gamma$ is the direct product of $\Gamma$ and $K_2$. If $\mathrm{Aut}(\mathrm{D}(\Gamma))=\mathrm{Aut}(\Gamma)\times\mathbb{Z}_2$ then $\Gamma$ is called stable; otherwise $\Gamma$…

A graph $\Gamma$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $\Gamma$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $\Gamma$ be a connected…

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\Gamma,G)$ where $\Gamma$ is 4-valent, connected and $G$-oriented ($G$-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive…

Let $\Gamma=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $\Gamma$ are integers, then we say that $\Gamma$ is an integral graph. A graph $\Gamma$ is determined by its spectrum if every graph cospectral to it…

In this paper we introduce a new domination problem strongly related to the following one recently proposed by Broe, Chartrand and Zhang. One says that a vertex $v$ of a graph $\Gamma$ labeled with an integer $\ell$ dominates the vertices…

The Gruenberg--Kegel graph (or the prime graph) $\Gamma(G)$ of a finite group $G$ is defined as follows. The vertex set of $\Gamma(G)$ is the set of all prime divisors of the order of $G$. Two distinct primes $r$ and $s$ regarded as…

A graph $G(V,E)$ is $\Gamma$-harmonious when there is an injection $f$ from $V$ to an Abelian group $\Gamma$ such that the induced edge labels defined as $w(xy)=f(x)+f(y)$ form a bijection from $E$ to $\Gamma$. We study $\Gamma$-harmonious…

An $H$-decomposition of a graph $\Gamma$ is a partition of its edge set into subgraphs isomorphic to $H$. A transitive decomposition is a special kind of $H$-decomposition that is highly symmetrical in the sense that the subgraphs (copies…

This paper initiates the investigation of the family of $(G,s)$-geodesic-transitive digraphs with $s\geq 2$. We first give a global analysis by providing a reduction result. Let $\Gamma$ be such a digraph and let $N$ be a normal subgroup of…

We prove that, given a finite graph $\Sigma$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $\Sigma$. Applying this result, we establish the existence of infinite families of…

A graph is said to be {\em half-arc-transitive} if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so called…