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In this paper, we study oriented bipartite graphs. In particular, we introduce "bitransitive" graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic…

Bidirected graphs are multigraphs where every edge has an independent direction at each end. In the paper, with an arbitrary bidirected graph we associate a non-negative integral quadratic form (called the incidence form of the graph), and…

We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not…

A semiregular permutation group on a set $\Ome$ is called {\em bi-regular} if it has two orbits. A classification is given of quasiprimitive permutation groups with a biregular dihedral subgroup. This is then used to characterize the family…

The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley…

A mixed dihedral group is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper, for each $n\geq 2$, we construct a…

It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilisers. However, beside a well known family of exceptional graphs, related to the lexicographic product of a cycle…

A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first step and the second step neighborhoods. In this paper, we first answer a question of A. Devillers, M. Giudici, C. H. Li and…

We extend the notion of an $H$-normal quotient digraph of an $H$-vertex-transitive digraph to that of an $H$-subnormal quotient digraph. Using these concepts, together with bipartite halves of bipartite digraphs, we show that, for each…

A graph $\G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $\Aut(\G)$, we say that $\G$ is {\em normal} with respect to…

Distance-regular graphs are a class of regualr graphs with pretty combinatorial symmetry. In 2007, Miklavi\v{c} and Poto\v{c}nik proposed the problem of charaterizing distance-regular Cayley graphs, which can be viewed as a natural…

A graph $X$ is 2-spanning cyclable if for any pair of distinct vertices $u$ and $v$ there is a 2-factor of $X$ consisting of two cycles such that $u$ and $v$ belong to distinct cycles. In this paper we examine the 2-spanning cyclability of…

A graph admitting an automorphism with two orbits of the same length is called a bicirculant. Recently, Jajcay et al. initiated the investigation of the edge-transitive bicirculants with the properties that one of the subgraphs induced by…

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…

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…

Let $G$ be a finite group. We consider the set of the irreducible complex characters of $G$, namely $Irr(G)$, and the related degree set $cd(G)=\{\chi(1) : \chi\in Irr(G)\}$. Let $\rho(G)$ be the set of all primes which divide some…

Kang and Park recently showed that every cubic (loopless) multigraph is incidence 6-choosable [On incidence choosability of cubic graphs. \emph{arXiv}, April 2018]. Equivalently, every bipartite graph obtained by subdividing once every edge…

A graph is called {\em arc-transitive} (or {\em symmetric}) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In…

We give a description of the cycle structure of the Heawood graph, $C_{14}$. In particular, we prove that the automorphism group of $C_{14}$ acts transitively on the set of $12$-cycles, Hamiltonian cycles, and disjoint pairs of $6$-cycles.…

We classify non-complete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of $2$-geodesics. We prove that either $\Gamma$ is 2-arc transitive or the valency $p$…