Title:

Vertexdecompositions of graphs

Certain partitions ("vertexdecompositions") of the vertices of a graph have the property that we can associate with them graphlike quotient structures ("generalised graphs"),and that we can define the adjacency matrix of a quotient in such a way that its eigenvalues and eigenvectors are closely related to those of the original graph. Chapters 1 and 2 give the basic definitions and results necessary to the rest of the thesis, and in addition Chapter 2 surveys most of the previous work in this field. Chapter 3 discusses vertexdecompositions of trees,and gives methods for finding the group and characteristic polynomial of a tree from its smallest quotient, Chapter 4 discusses vertexdecompositions of regular graphs into two classes, relating the existence of such decompositions to the possession of integer eigenvalues,and to switching classes of graphs. Chapter 5 considers graphs for which a quotient may possess all the eigenvalues of the graph from which it is derived, and demonstrates that for a particular class ("singletonregular" graphs), which includes vertextransitive graphs,it is possible to find not only the eigenvalues but also some of their multiplicities from the adjacency matrix of the quotient. Using this result, Chapter 6 and Appendix 1 are devoted to the construction of all possible quotients (with certain properties) of certain types of singletonregular graph, and in Chapter 7 we decide for each quotient whether or not there are any graphs with the desired properties»sometimes utilising arguments concerning the primitivity of the action of the automorphism group or the necessity for a graph to be a covering of a transitive graph. Exhaustive lists of graphs with the given properties are given, including a list of all symmetric trivalent graphs on < /40 vertices. Appendix 2 consists of a single proof establishing the uniqueness of the (3,12)cage by the use of a computer program, a result used in Chapter 7.
