Related papers: Partial difference sets and amorphic Cayley scheme…

We characterize connected tetravalent graphs $\Gamma$ which admit groups $M<H$ of automorphisms such that $\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive. Examples for each case are constructed, including a counter-example to a…

We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called `Taylor--Wiles hypothesis'. We apply this to the problem of the…

We complete the description of automorphism groups of all nonsplit extensions of elementary abelian $2$-groups by $PSL_2(q)$, with $q$ odd, for an irreducible induced action. An application of this result to the theory of $\pi$-submaximal…

We formulate a notion of the quantum automorphism group of a $2$-graph. After some preliminary computations, we define quantum isomorphism between a pair of $2$-graphs. We produce a `non-trivial' example of a pair of $2$-graphs that are not…

In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As…

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…

We characterise connected cubic graphs admitting a vertex- transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a…

We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such…

Let $X$ be a countable discrete Abelian group containing no elements of order 2, $\alpha$ be an automorphism of $X$, $\xi_1$ and $\xi_2$ be independent random variables with values in the group $X$ and distributions $\mu_1$ and $\mu_2$. The…

Let $G_n=\mathbb{Z}_n\times \mathbb{Z}_n$ for $n\geq 4$ and $S=\{(i,0),(0,i),(i,i): 1\leq i \leq n-1\}\subset G_n$. Define $\Gamma(n)$ to be the Cayley graph of $G_n$ with respect to the connecting set $S$. It is known that $\Gamma(n)$ is a…

In this paper, we study the metric dimension of Cayley graphs. Specially, we present a complete characterization of Cayley graphs on Abelian groups whose metric dimension is two.

In this work we are interested in the general problem of the determination of the normed division algebras. Our fundamental results are obtained in the particular subclass of those 8-dimensional quadratic flexible real division algebras. We…

In this paper, we explore non-abelian extensions of relative Rota-Baxter Lie algebras and classify the non-abelian extensions by introducing the non-abelian second cohomology group. We also study the inducibility of a pair of automorphisms…

It is known that the canonical double cover of any connected nonbipartite graph have an automorphism group of the form $H \rtimes \mathbb{Z}_2$, where $H$ is the set of automorphism which preserve bipartite parts. We construct connected…

In 1988, Duval introduced the concept of directed strongly regular graphs, which can be viewed as a directed graph version of strongly regular graphs. Such directed graphs have similar structural and algebraic properties to strongly regular…

We give two new constructions of almost difference sets. The first is a generic construction of $(q^{2}(q+1),q(q^{2}-1),q(q^{2}-q-1),q^{2}-1)$ almost difference sets in certain groups of order $q^{2}(q+1)$ ($q$ is an odd prime power) having…

This paper investigates the enumeration of Cayley digraphs, focusing on counting Cayley digraphs on dihedral groups up to CI-isomorphism. By leveraging the Cauchy-Frobenius Lemma and properties of automorphisms, we derive an explicit…

The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel…

In this paper, we describe the automorphism group of semidirect product of two groups that fixes the non-normal subgroup of it. We have computed these automorphisms for the non-abelian metacyclic $p$-group and non-abelian $p$-groups $(p\ge…

Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square type graphs and whose underlying set is a…