We introduce the concept of a pentagonal geometry as a generalization of the pentagon and the Desargues configuration, in the same vein that the generalized polygons share the fundamental properties of ordinary polygons. In short, a pentagonal geometry is a regular partial linear space in which for all points x, the points not collinear with the point x, form a line. We compute bounds on their parameters, give some constructions, obtain some nonexistence results for seemingly feasible parameters and suggest a cryptographic application related to identifying codes of partial linear spaces.

No abstract

The distinguishing number of G Sym(Ω) is the smallest size of a partition of Ω such that only the identity of G fixes all the parts of the partition. Extending earlier results of Cameron, Neumann, Saxl and Seress on the distinguishing number of finite primitive groups, we show that all imprimitive quasiprimitive groups have distinguishing number two, and all non-quasiprimitive semiprimitive groups have distinguishing number two, except for GL(2, 3) acting on the eight non-zero vectors of F 2 3 , which has distinguishing number three.

A partial linear space is a non-empty set of points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points. Graphs and linear spaces are particular cases of partial linear spaces. A partial linear space which is not a graph or a linear space is called proper. In this paper, we give a complete classification of all finite proper partial linear spaces admitting a primitive rank 3 automorphism group of almost simple type.

A 2-geodesic in a graph is a vertex triple (u, v, w) such that v is adjacent to both u and w and u, w are not adjacent. We study non-complete graphs Γ in which, for each vertex u, all 2-geodesics with initial vertex u are equivalent under the subgroup of graph automorphisms fixing u. We call such graphs locally 2-geodesic transitive, and show that the subgraph [Γ (u)] induced on the set of vertices of Γ adjacent to u is either (i) a connected graph of diameter 2, or (ii) a union mK r of m 2 copies of a complete graph K r with r 1. This suggests studying locally 2-geodesic transitive graphs according to the structure of the subgraphs [Γ (u)]. We investigate the family F (m, r) of connected graphs Γ such that [Γ (u)] ∼ = mK r for each vertex u, and for fixed m 2, r 1. We show that each Γ ∈ F (m, r) is the point graph of a partial linear space S of order (m, r + 1) which has no triangles (and 2-geodesic transitivity of Γ corresponds to natural strong symmetry properties of S). Conversely, each S with these properties has point graph in F (m, r), and a natural duality on partial linear spaces induces a bijection F (m, r) → F (r + 1, m − 1).

The automorphism group of a flag-transitive 6-(v, k, 2) design is a 3-homogeneous permutation group. Therefore, using the classification theorem of 3-homogeneous permutation groups, the classification of flag-transitive 6-(v, k,2) designs can be discussed. In this paper, by analyzing the combination quantity relation of 6-(v, k, 2) design and the characteristics of 3-homogeneous permutation groups, it is proved that: there are no 6-(v, k, 2) designs D admitting a flag transitive group G ≤ Aut (D) of automorphisms.

Abstract. We give a unified approach to analysing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s-arc transitive graphs. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s ≥ 2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph, or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups.

The subdivision graph S(Σ) of a graph Σ is obtained from Σ by 'adding a vertex' in the middle of every edge of Σ. Various symmetry properties of S(Σ) are studied. We prove that, for a connected graph Σ, S(Σ) is locally s-arc transitive if and only if Σ is ⌈ s+1 2 ⌉-arc transitive. The diameter of S(Σ) is 2d + δ, where Σ has diameter d and 0 δ 2, and local s-distance transitivity of S(Σ) is defined for 1 s 2d + δ. In the general case where s 2d − 1 we prove that S(Σ) is locally s-distance transitive if and only if Σ is ⌈ s+1 2 ⌉-arc transitive. For the remaining values of s, namely 2d s 2d + δ, we classify the graphs Σ for which S(Σ) is locally s-distance transitive in the cases, s 5 and s 15 + δ. The cases max{2d, 6} s min{2d + δ, 14 + δ} remain open.