Let (P, L, *) be a near polygon having 8 + 1 points per line, s > 1, and suppose k is a field. Let Vk be the k-vector space with basis {vp | p € P}. Then the subspace generated by the vectors v1 = Lp*l vp, where l 6 L, has codimension at least 2 in Vk. This observation is used in two ways. First we derive the existence of certain diagram geometries with flag transitive automorphism group, and s...

An action of a group on a set is called k-transitive if it is transitive on ordered k-tuples and highly transitive if it is k-transitive for every k. We show that for n ≥ 4 the group Out(Fn) = Aut(Fn)/ Inn(Fn) admits a faithful highly transitive action on a countable set.

The publication of knowledge graphs on the Web in form RDF datasets, and subsequent integration such are both essential to idea Linked Open Data. Combining can result undesirable graph structures even logical inconsistencies. Refinement methods that detect repair therefore crucial importance. Existing refinement for often domain-specific, limited single relations (e.g. owl:sameAs), or scale. We...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l) set of vertices. A k-kernel is a (k, k − 1)-kernel. A digraph D is transitive if (u, v), (v, w) ∈ A(D) implies that (u,w) ∈ A(D). This co...

A subgroup G of automorphisms of a graph X is said to be 1 2-transitive if it is vertex and edge but not arc-transitive. The graph X is said to be 1 2-transitive if Aut X is 1 2-transitive. The graph X is called one-regular if Aut X acts regularly on the set arcs of X. The interplay of three diierent concepts of maps, one-regular graphs and 1 2-transitive group actions on graphs of valency 4 is...

The theory of vertex-transitive graphs has developed in parallel with the theory of transitive permutation groups. In this chapter we explore some of the ways the two theories have influenced each other. On the one hand each finite transitive permutation group corresponds to several vertex-transitive graphs, namely the generalised orbital graphs which we shall discuss below. On the other hand, ...