Related papers: Higher-dimensional grid-imprimitive block-transiti…

A code $C$ in the Hamming graph $\varGamma=H(m,q)$ is $2\it{\text{-neighbour-transitive}}$ if ${\rm Aut}(C)$ acts transitively on each of $C=C_0$, $C_1$ and $C_2$, the first three parts of the distance partition of $V\varGamma$ with respect…

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point-imprimitive group action, for a given value of gcd(k,r), where k is the line size and r is the…

The Clifford group is the quotient of the normalizer of the Weyl-Heisenberg group in dimension $d$ by its centre. We prove that when $d$ is not prime the Clifford group is not a group unitary $2$-design. Furthermore, we prove that the…

It is well-known that every sharply 2-transitive group of characteristic 3 splits. Here we construct the first examples of non-split sharply 2-transitive groups in odd positive characteristic $p$, for sufficiently large primes $p$.…

In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information…

If totally periodic points are dense in a subshift $X$, its automorphism group is residually finite. We show a weak converse: if periodic points are not dense in a subshift $X$, then the automorphism group of $X \times Y$ is not residually…

A notion of $t$-designs in the symmetric group on $n$ letters was introduced by Godsil in 1988. In particular $t$-transitive sets of permutations form a $t$-design. We derive special lower bounds for $t=1$ and $t=2$ by a power moment…

In this article, we study $2$-designs with $\gcd(r,\lambda)=1$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type. We obtain four infinite families of such designs and…

In this article, we study symmetric designs admitting flag-transitive, point-imprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting…

The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset \Gamma such…

A connected graph $\Gamma=(V,E)$ of valency at least $3$ is called a basic $2$-arc-transitive graph if its full automorphism group has a subgroup $G$ with the following properties: (i) $G$ acts transitively on the set of $2$-arcs of…

In this article, we study symmetric $(v, k, \lambda)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is a projective special unitary group of dimension at most five. We, in particular, determine…

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…

Block graphs are graphs in which every block (biconnected component) is a clique. A graph $G=(V,E)$ is said to be an (unpartitioned) $k$-probe block graph if there exist $k$ independent sets $N_i\subseteq V$, $1\le i\le k$, such that the…

Among the properties of homogeneity of incidence structures flag-transitivity obviously is a particularly important and natural one. Consequently, in the last decades also flag-transitive Steiner tdesigns (i.e. flag-transitive t-(v,k,1)…

This paper is devoted to the classification of all flag-transitive point-primitive non-trivial $2$-$(v, k, \lambda)$ designs with the alternating group $A_n$($n \le 10$) as the socle of their automorphism groups, and 87 different designs…

This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that…

In this paper new Steiner systems $S(2,6,111)$, $S(2,6,121)$, $S(2,6,126)$, $S(2,7,169)$, $S(2,7,175)$ and possibly others with point-transitive (commutative except $S(2,6,111)$ case) automorphism groups are introduced.

A graph $\Gamma$ is called $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of $G$-symmetric graphs $\Gamma$ with $V(\Gamma)$ admitting…

A group divisible design $\mbox{GDD}(m,n;\lambda_1,\lambda_2)$, is an ordered pair $(V, \cal{B})$ where $V$ is an $(m+n)$-set of symbols while $\cal{B}$ is a collection of $3$-subsets (called blocks) of $V$ satisfying the following…