A formula for the number of weak endomorphisms on paths

A weak endomorphisms of a graph is a mapping on the vertex set of the graph which preserves or contracts edges. In this paper we provide a formula to determine the cardinalities of weak endomorphism monoids of finite undirected paths

Cathodoluminescence and TEM investigations of structural and optical properties of AlGaN on epitaxial laterally overgrown AlN/sapphire templates

Surface steps as high as 15 nm on up to 10 μm thick AlN layers grown on patterned AlN/sapphire templates play a major role for the structural and optical properties of AlxGa1−xN layers with x ≥ 0.5 grown subsequently by metalorganic vapour phase epitaxy. The higher the Ga content in these layers is, the stronger is the influence of the surface morphology on their properties. For x = 0.5 not only periodic inhomogeneities in the Al content due to growth of Ga-rich facets are observed by cathodoluminescence, but these facets give rise to additional dislocation formation as discovered by annular dark-field scanning transmission electron microscopy. For AlxGa1−xN layers with x = 0.8 the difference in Al content between facets and surrounding material is much smaller. Therefore, the threading dislocation density (TDD) is only defined by the TDD in the underlying epitaxially laterally overgrown (ELO) AlN layer. This way high quality Al0.8Ga0.2N with a thickness up to 1.5 μm and a TDD ≤ 5x108 cm−2 was obtained

Characterization of finite simple semigroup digraphs

This paper characterizes directed graphs which are Cayley graphs of finite simple semigroups, i.e. of a subspecies of completely regular semigroups. Moreover we investigate the structure of Cayley graphs of finite simple semigroups with a one-element connection set. We introduce the conditions for which they are isomorphic and connected

The endomorphisms monoids of graphs of order n with a minimum degree n − 3

We characterize the endomorphism monoids, End(G), of the generalized graphs G of order n with a minimum degree n − 3. Criteria for regularity, orthodoxy and complete regularity of those monoids based on the structure of G are given

The Endomorphism Monoids of (n − 3)-regular Graphs of Order n

This paper is motivated by the result of W. Li, he presents an infinite family of graphs - complements of cycles - which possess a regular monoid. We show that these regular monoids are completely regular. Furthermore, we characterize the regular, orthodox and completely regular endomorphisms of the join of complements of cycles, i.e. (n−3)-regular graph of order n

Covers of acts over monoids II

In 1981 Edgar Enochs conjectured that every module has a flat cover and finally proved this in 2001. Since then a great deal of effort has been spent on studying different types of covers, for example injective and torsion free covers. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids but their definition of cover was slightly different from that of Enochs. Recently, Bailey and Renshaw produced some preliminary results on the `other' type of cover and it is this work that is extended in this paper. We consider free, divisible, torsion free and injective covers and demonstrate that in some cases the results are quite different from the module case

Faster fixed-parameter tractable algorithms for matching and packing problems. In:

Abstract We obtain faster algorithms for problems such as r-dimensional matching and r-set packing when the size k of the solution is considered a parameter. We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). This technique lets us combine Alon, Yuster and Zwick's colorcoding technique with dynamic programming to obtain faster fixed-parameter algo- rithms for these problems. Our algorithms run in time O(n + 2 O(k) ), an improvement over previous algorithms for some of these problems running in time O(n + k O(k) ). The flexibility of our approach allows tuning of algorithms to obtain smaller constants in the exponent