A cluster of cycles (or (r; q)-polycycle) is a simple planar 2{connected nite or countable graph G of girth r and maximal vertex-degree q, which admits an (r; q)-polycyclic realization P(G) on the plane. An (r; q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q, (ii) all interior faces (denote their number by p r) are combinatorial r-g...

In the extension of irregularity indices, Abdo et. al. [1] defined the total irregu-larity of a graph G = (V, E) as irrt(G) = 21 Pu,v∈V (G) du − dv, where du denotesthe vertex degree of a vertex u ∈ V (G). In this paper, we investigate the totalirregularity of trees with bounded maximal degree Δ and state integer linear pro-gramming problem which gives standard information about extremal trees a...

A Turing degree a is said to isolate another degree b in the r.e. degrees if and only if a < b and there is no r.e. degree c with a < c < b. In this case b is called an isolated degree and a an isolating degree. There are of course analogous deenitions for any subsets A and B of !. If a isolates b, it follows from the Sacks density theorem that at least one of the two must fail to be r.e. When ...

We propose a reformulation of general relativity by making the Abelian decomposition of Einstein’s theory. Based on the view that Einstein’s theory can be interpreted as a gauge theory of Lorentz group, we decompose the Einstein’s gravitational connection (the gauge potential of Lorentz group Γμ) into the restricted connection made of the potential of the maximal Abelian subgroup H of Lorentz g...

The purpose of this paper is to develop a transverse notion of Lusternik{Schnirelmann category in the eld of foliations. Our transverse category, denoted cat\j (M;F), is an invariant of the foliated homotopy type which is nite on compact manifolds. It coincides with the classical notion when the foliation is by points. We prove that for any foliated manifold catM catL cat\j (M;F), where L is a ...

Apart from hyperovals and their duals there are only three classes of maximal arcs known in Desarguesian projective planes. Two classes are due to J. A. Thas and one to R. H. F. Denniston. In this paper collineation stabiliser and isomorphism problems for those maximal arcs in Desarguesian projective planes are examined. The full collineation stabilisers of the known maximal arcs are calculated...