Let PG(r, q) be the projective space of dimension r over GF (q). A k–cap K̄ in PG(r, q) is a set of k points, no three of which are collinear [10], and a k–cap is said to be complete if it is maximal with respect to set–theoretic inclusion. The maximum value of k for which there is known to exist a k–cap in PG(r, q) is denoted by m2(r, q). Some known bounds for m2(r, q) are given below. Suppose ...

We present two new compact transitive closure representations. The rst uses intervals and the second chains to store the closure. Both representations are based on previous methods designed for acyclic graphs. The new representations are applicable to all kinds of graphs, and can be eeciently constructed during a single traversal of the input graph. We compared experimentally the average size o...

Given that A is longer than B, and that B is longer than C, even 5-year-old children can infer that A is longer than C. Theories of reasoning based on formal rules of inference invoke simple axioms ("meaning postulates") to capture such transitive inferences. An alternative theory proposes instead that reasoners construct mental models of the situation described by the premises in order to draw...

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, ...