Graph Relations and Constrained Homomorphism Partial Orders

We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives from relations between graphs and is related to multihomomorphisms. This gives a generalization of surjective homomorphisms and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time. The theory of the graph homomorphism order is well developed, and from it we consider analogous notions defined for orders induced by constrained homomorphisms. We identify corresponding cores, prove or disprove universality, characterize gaps and dualities. We give a new and significantly easier proof of the universality of the homomorphism order by showing that even the class of oriented cycles is universal. We provide a systematic approach to simplify the proofs of several earlier results in this area. We explore in greater detail locally injective homomorphisms on connected graphs, characterize gaps and show universality. We also prove that for every $d\geq 3$ the homomorphism order on the class of line graphs of graphs with maximum degree $d$ is universal

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Pollination ecology and the functional significance of unusual floral traits in two South African stapeliads.

Master of Science in Ecology. University of KwaZulu-Natal, Durban 2017.Carrion and dung mimicking plants often exhibit unusual floral traits which are believed to attract necro- and coprophagous insects as pollinators. Our understanding of these unusual traits and their functions is very limited. Stapeliads (Apocynaceae: Asclepiadoideae: Stapeliinae) are a monophyletic group of some 400 species of stem-succulent plants, many of which emit foul odours and exhibit unusual morphological traits that have anecdotally been assumed to represent adaptations to enhance the flowers’ resemblance to carrion or dung. This study looked at the pollination biology of two stapeliads, Orbea variegata and Stapelia hirsuta var. hirsuta, and explored the functional significance of some of the floral traits commonly associated with carrion or dung mimicking flowers. Further, odours emitted by both species were compared to the odours of putative models to explore the chemical basis for the assumed mimicry. Orbea variegata attracted flies from the families Muscidae, Calliphoridae and Sarcophagidae (at sites near Scarborough and Clifton, Western Cape) and individuals from each of these families were found carrying pollinia. The scent of O. variegata flowers was found to be dominated by dimethyl disulphide, dimethyl trisulphide as well as phenol. The presence of both these compounds suggests mimicry of both carrion and dung, although an ANOSIM analysis indicated that the odour of O. variegata shared more similarities with dung. This suggests that O. variegata is fairly generalist and explains the attraction of various flies that are associated with carrion or faeces by the flowers of this species. In experiments testing the importance of black versus yellow colouring and the importance of patterning, flies were found to prefer black coloured models in the presence of O. variegata odour, whereas the presence and size of blotching on the corolla lobes had no significant effect on fly visits. The colours of the black blotching and yellow of the corolla lobes showed minimal chromatic contrast when interpreted using the Troje (1993) fly vision model, although background rocks showed chromatic contrast, suggesting flies can distinguish between the flowers and the background. In an experiment testing the importance of odour for attracting flies, significantly fewer flies were able to locate concealed flowers compared to visible flowers, suggesting an important role for visual cues for flies to locate the odour source. Stapelia hirsuta var. hirsuta was found to exhibit two floral colour morphs at Swellendam (Western Cape). The yellow morph was rarer than the maroon morph. These flowers attracted flies belonging to the Muscidae, Calliphoridae and Sarcophagidae families, although only Calliphoridae and Sarcophagidae were found to carry pollinia. The odour composition of these two morphs differed slightly, where the odour of the maroon morph was dominated by dimethyl disulphide, dimethyl trisulphide and p-cresol and the yellow morph was dominated by dimethyl disulphide, dimethyl trisulphide and limonene. The ANOSIM analysis of odours emitted by S. hirsuta var. hirsuta in relation to that of various fly oviposition substrates suggested that these flowers are dung mimics rather than carrion mimics, although the presence of sulphides suggests possible mimicry of both. The yellow morph had higher fly visitation rates than the maroon morph. In experiments testing the role of floral trichomes, the removal of floral trichomes significantly decreased the visitation rates to the flowers, as well as the amount of time visitors spend on the flowers. Again, visual cues were shown to be of importance, as visible flowers received more visits than concealed flowers. Analysis of colours of different floral morphs, using the Troje (1993) fly vision model, suggests that flies cannot perceive chromatic colour differences between morphs. In these studies, I have shown that O. variegata and S. hirsuta var. hirsuta are visited and pollinated by carrion associated flies, and the flowers emit odours associated with both carrion and dung. This work sheds light on some of the floral features that are often associated with carrion and dung mimicry by flowers and the roles they play in the attraction of flies

Basic Neutrosophic Algebraic Structures and their Application to Fuzzy and Neutrosophic Models

The involvement of uncertainty of varying degrees when the total of the membership degree exceeds one or less than one, then the newer mathematical paradigm shift, Fuzzy Theory proves appropriate. For the past two or more decades, Fuzzy Theory has become the potent tool to study and analyze uncertainty involved in all problems. But, many real-world problems also abound with the concept of indeterminacy. In this book, the new, powerful tool of neutrosophy that deals with indeterminacy is utilized. Innovative neutrosophic models are described. The theory of neutrosophic graphs is introduced and applied to fuzzy and neutrosophic models. This book is organized into four chapters. In Chapter One we introduce some of the basic neutrosophic algebraic structures essential for the further development of the other chapters. Chapter Two recalls basic graph theory definitions and results which has interested us and for which we give the neutrosophic analogues. In this chapter we give the application of graphs in fuzzy models. An entire section is devoted for this purpose. Chapter Three introduces many new neutrosophic concepts in graphs and applies it to the case of neutrosophic cognitive maps and neutrosophic relational maps. The last section of this chapter clearly illustrates how the neutrosophic graphs are utilized in the neutrosophic models. The final chapter gives some problems about neutrosophic graphs which will make one understand this new subject.Comment: 149 pages, 130 figure

Simplicial Models for the Epistemic Logic of Faulty Agents

In recent years, several authors have been investigating simplicial models, a model of epistemic logic based on higher-dimensional structures called simplicial complexes. In the original formulation, simplicial models were always assumed to be pure, meaning that all worlds have the same dimension. This is equivalent to the standard S5n semantics of epistemic logic, based on Kripke models. By removing the assumption that models must be pure, we can go beyond the usual Kripke semantics and study epistemic logics where the number of agents participating in a world can vary. This approach has been developed in a number of papers, with applications in fault-tolerant distributed computing where processes may crash during the execution of a system. A difficulty that arises is that subtle design choices in the definition of impure simplicial models can result in different axioms of the resulting logic. In this paper, we classify those design choices systematically, and axiomatize the corresponding logics. We illustrate them via distributed computing examples of synchronous systems where processes may crash

Algebraic number-theoretic properties of graph and matroid polynomials

PhDThis thesis is an investigation into the algebraic number-theoretical properties of certain polynomial invariants of graphs and matroids. The bulk of the work concerns chromatic polynomials of graphs, and was motivated by two conjectures proposed during a 2008 Newton Institute workshop on combinatorics and statistical mechanics. The first of these predicts that, given any algebraic integer, there is some natural number such that the sum of the two is the zero of a chromatic polynomial (chromatic root); the second that every positive integer multiple of a chromatic root is also a chromatic root. We compute general formulae for the chromatic polynomials of two large families of graphs, and use these to provide partial proofs of each of these conjectures. We also investigate certain correspondences between the abstract structure of graphs and the splitting fields of their chromatic polynomials. The final chapter concerns the much more general multivariate Tutte polynomials—or Potts model partition functions—of matroids. We give three separate proofs that the Galois group of every such polynomial is a direct product of symmetric groups, and conjecture that an analogous result holds for the classical bivariate Tutte polynomial

Colour and Naming in Healthy and Aphasic People

Abstract The purpose of this study was to create a paradigm suitable for people with aphasia and healthy subjects to evaluate the influence of colour on naming pictures of objects. We designed a completely new stimulus set based on images of 140 common real objects that were inspired by the Snodgrass and Vanderwart picture set (1980). We were especially interested whether there is a difference in performance between the aphasic patients and the group of healthy controls. Adding chromatic information to pictures of objects shows only a small effect in verification and categorisation tasks. However, when observers are required to name objects, colour speeds performance and enhances accuracy (Rossion & Pourtois, 2004). The present study contrasts two different claims as to why colour may benefit object naming. The first is that colour simply aids the segmentation of the object from its background (Wichmann et al., 2002). The second is that colour may help to elicit a wider range of associations with the object, thereby enhancing lexical access (Bisiach, 1966). To distinguish between these processes an equal number of pictures containing high and low colour diagnostic objects were presented against either fractal noise or uniform backgrounds in a naming task to aphasic subjects with anomia and to healthy controls. Performance for chromatic stimuli was compared with that for monochrome stimuli equated in luminance. Results show that colour facilitates naming significantly in both subject groups and there was no significant difference between objects with high or low colour diagnostic values. We also found that object segmentation and the lexical access seem to occur in parallel processes, rather than in an additive way