arXiv subjects
Publications and source records attributed to Shenglin Zhou.
The classification of the $2$-designs with $\lambda=2$ admitting a flag-transitive automorphism groups with socle $PSL(2,q)$ is completed by settling the two open cases in \cite{ABDT}. The result is achieved by using conics and hyperovals of $PG(2,q)$.
A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its vertex set admits a nontrivial $G$-invariant partition ${\cal B}$, and let ${\cal D}(\Gamma, {\cal B})$ be the incidence structure with point set ${\cal B}$ and blocks $\{B\} \cup \Gamma_{\cal B}(\alpha)$, for $B \in {\cal B}$ and $\alpha \in B$, where $\Gamma_{\cal B}(\alpha)$ is the set of blocks of ${\cal B}$ containing at least one neighbour of $\alpha$ in $\Gamma$. In this paper we classify all $G$-symmetric graphs $\Gamma$ such that $\Gamma_{\cal B}(\alpha) \ne \Gamma_{\cal B}(\beta)$ for distinct $\alpha, \beta \in B$, the quotient graph of $\Gamma$ with respect to ${\cal B}$ is a complete graph, and ${\cal D}(\Gamma, {\cal B})$ is isomorphic to the complement of a $(G, 2)$-point-transitive linear space.
In this paper we show that a flag-transitive automorphism group $G$ of a non-trivial $2$-$(v,k,λ)$ design with $λ\geq (r, λ)^2$ is not of product action type. In conclusion, $G$ is a primitive group of affine or almost simple type.
Let $G$ be a flag-transitive automorphism group of a $(v,k,λ)$ symmetric design $\mathcal{D}$ with $k>λ(λ-2)$. O'Reilly Regueiro proved that if $G$ is point-imprimitive, then $\mathcal{D}$ has parameters $(v,k,λ)=(λ^2(λ+2),λ(λ+1),λ)$. In the present paper, we consider the case that $G$ is point-primitive. By applying the O'Nan-Scott Theorem, we prove that $G$ must be of affine type or almost simple type.
In this paper, we present a classification of $2$-designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,λ)=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q = p^{d}$ points with $p$ prime and $G$ is a subgroup of $AΓL_{1}(q)$.
This paper determined all pairs $(\mathcal{D},G)$ where $\mathcal{D}$ is a non-symmetric 2-$(v,k,λ)$ design with $(r,λ)=1$ and $G$ is the almost simple flag-transitive automorphism group of $\mathcal{D}$ with an exceptional socle of Lie type. We prove that if $T\trianglelefteq G\leq Aut(T)$ where $T$ is an exceptional group of Lie type, then $T$ must be the Ree group or Suzuki group, and there are five classes of non-isomorphic designs $\mathcal{D}$.
Let $G=(V,E)$ be a simple connected graph. A set of vertices $S\subseteq V$ is said to be a dominating set if for any vertex in $V\setminus S$ is adjacent to at least one vertex in $S$. The domination number $γ(G)$ of $G$ is the minimum cardinality among all such sets. In this paper, we obtain some results on the domination number of the incidence graphs of combinatorial designs. In particular, we prove a conjecture and disprove another conjecture in a recent paper by Goldberg, Rajendraprasad and Mathew. We also prove a third conjecture by the same authors for block-transitive symmetric designs.
The purpose of this paper is to classify all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a non-trivial $2$-$(v, k, 2)$ design, and $G\leq Aut(\mathcal{D})$ acts transitively on the set of blocks of $\mathcal{D}$ and primitively on the set of points of $\mathcal{D}$ with sporadic socle. We prove that there exists only one such pair $(\mathcal{D}, G)$ in which $\mathcal{D}$ is a $2$-$(176,8,2)$ design and $G=HS$, the Higman-Sims simple group.
This paper studies flag-transitive point-primitive non-symmetric $2$-($v,k,2$) designs. We prove that if $\mathcal{D}$ is a non-trivial non-symmetric $2$-$(v,k,2)$ design admitting a flag-transitive point-primitive automorphism group $G$ with $Soc(G)=A_{n}$ for $n\geq5$, then $\mathcal{D}$ is a $2$-$(6,3,2)$ or $2$-$(10,4,2)$ design.
Let $\mathcal D$ be a nontrivial symmetric $(v,k,λ)$ design, and $G$ be a subgroup of the full automorphism group of $\mathcal D$. In this paper we prove that if $G$ acts flag-transitively, point-primitively on $\mathcal D$ and $Soc(G)= PSL(2,q)$, then D has parameters $(7, 3, 1)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 3)$ or $(15, 8, 4)$.
This paper is a further contribution to the classification of line-transitive finite linear spaces. We prove that if S is a non-trivial finite linear space with the Fang-Li parameter gcd(k,r) is 9 or 10, the automorphism group G of S is line-transitive and point-imprimitive, then S is the Desarguesian projective plane PG(2,9).
In this paper we complete a classification of finite linear spaces $\cS$ with line size at most 12 admitting a line-transitive point-imprimitive subgroup of automorphisms. The examples are the Desarguesian projective planes of orders $4,7, 9$ and 11, two designs on 91 points with line size 6, and 467 designs on 729 points with line size 8.
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 number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k,r) at most 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of algorithms that, for gcd(k,r) up to a given maximum value, return a list of candidate parameter values and candidate groups. We examine in detail each of the possibilities returned by these algorithms for gcd(k,r) at most 8, and complete the classification in this case.