Functions on graphs and some generalizations

Abstract

The underlying theme of this thesis is the investigation of some functions defined on graphs. In one chapter results are obtained for a particular type of function. In another a known result on expressing graph functions multiplicatively is established by other methods which suggest certain generalizations of graphs; these then enable us to see how much the properties depend on graphs per se, and how much on the wider nature of the generalizations. Chapter 1 introduces our basic terminology and notation, and contains the acknowledgements. Chapter 2 establishes the multiplicative expansion of a graph function by two new methods. Chapter 3 is simply a survey of some combinatorial structures and the definition of some new ones. The 'old' ones are introduced for the purpose of comparison with the new ones, which in turn afford useful generalizations of graphs which are utilised in later chapters. Chapter investigates interaction models on graphs that possess a certain additive property; this is then extended to hypergraphs and related to rank polynomials. Chapter 5 pursues the notion of 'avitoids', defined in chapter 3. A rank function is defined on them and some of its properties studied. Chapter 6 is concerned with the idea of molecules', also defined in chapter 3* A simple way to evaluate, in general circumstances, an important matrix that emerged in chapter 2 is presented. Chapter 7 concludes the thesis with comments on connected and resultant items. More detailed summaries are provided at the beginning of each chapter.