Chapter I begins with a brief history of the topic of greatest common subgraphs. Then we provide a summaiy of the work done on some variations of greatest common subgraphs. Finally, in this chapter we present results previously obtained on greatest common divisors and least common multiples of graphs. In Chapter II the concepts of prime graphs, prime divisors of graphs, and primeconnected graphs are presented. We show the existence of prime trees of any odd size and the existence of prime-connected trees that are not prime having any odd composite size. Then the number of prime divisors in a graph is studied. Finally, we present several results involving the existence of graphs whose size satisfies some prescribed condition and which contains a specified number of prime divisors. Chapter III presents properties of greatest common divisors and least common multiples of graphs. Then graphs with a prescribed number of greatest common divisors or least common multiples are studied. In Chapter IV we study the sizes of greatest common divisors and least common multiples of specified graphs. We find the sizes of greatest common divisors and least common multiples of stars and that of stripes. Then the size of greatest common divisors and least common multiples of paths and complete graphs are investigated. In particular, the size of least common multiples of paths versus K3 or K4 are determined. Then we present the greatest common divisor index of a graph and we determine this parameter for several classes of graphs. iii In Chapter V greatest common divisors and least common multiples of digraphs are introduced. The existence of least common mutliples of two stars is established, and the size of a least common multiple is found for several pairs of stars. Finally, we present the concept of greatest common divisor index of a digraph and determine it for several classes of digraphs. iv / Mathematical Sciences / Ph. D. (Mathematical sciences)

Chapter 0 details the notation and terminology used. Chapter 1 introduces the usual linear algebra over GF2 of edge space E and its orthogonal subspaces Z (cycle space) and Z* (cut space). "Reduced vectors" are defined as elements of the quotient space E/Z*. Reduced vectors of edges give a simple way of characterising edges that are bridges (their reduced vector is null) or 2-edge cuts (their vectors are equal), and also of spanning trees (the edges outside the tree are a basis) and form to the best of my knowledge a new approach. They are also useful in later chapters to describe Tait colorings, as well as cycle double covers. Perhaps the most important property of E/Z* is the Unique graph theorem: unlike in E, a list of which reduced vectors are edges uniquely determines graph structure (if edge connectivity is high enough; that covers certain “solid” components every trivalent graph can be decomposed into). Chapter 2 gives a brief intoduction to graph embeddings and planar graphs. Chapter 3 deals specifically with trivalent graphs, listing some of the ways in which they are different from graphs in general. Results here include two versions of Bipolar growth theorem which can be used for constructive proofs, and (after defining “halftrees” and a “flipping” operation between them) a theorem enumerating the set C\(_n\) of halftrees of a given size, the "Caterpillar theorem" showing C\(_n\) is connected by flipping, and the "Butterfly theorem" derived from it. Graphs referred to here as "solid" are shown to play an important structural role. Chapter 4 deals with the 4-coloring theorem. The first half shows the older results in a unified light using edge spaces over GF4. The second half applies methods from coding theory to this. The 4-color theorem is shown to be equivalent to a variety of statements about cycle-shaped words in codes over GF4 or GF3, many of them tantalisingly simple to state (but not, as yet, to prove). Chapter 5 deals with what has been variously called polyhedral decompositions and (specifically for those using cycles) cycle double covers, as in the cycle double cover conjecture. The more general concept is referred to as a "map" in this paper, and identified with what is termed here "cisness structures", which is a new approach. There is also a simpler proof of a theorem by Szekeres. Links with the subject of the previous chapter are identified, and some approaches towards proving the conjecture suggested. Several planned appendices were left out of the version submitted for examination because they would make the thesis too big, and/or were not finished. Of the ones that remain, appendix H (on embedding infinite 4- and 3-valent trees X and Y in the hyperbolic plane) now seems disjointed from the body of the text (a planned appendix dealt with colorings of finite graphs as the images of homomorphisms from embeddings of Y). Appendix B enumerates cycle maps (cycle double covers) on a number of small graphs while appendix D investigates the dimension of the instersection of Z and Z*.

A theorem of Talbot, stated in terms of graph theory, is as follows: Let G be a cycle on n vertices raised to the power k > 1. If A is a family of intersecting independent r-subsets of the vertices of G where r > 1 and n > (k: + 1)r then [A] < (n-kr-1 / r-1).

Chapter I begins with a brief history of the topic of greatest common subgraphs. Then we provide a summaiy of the work done on some variations of greatest common subgraphs. Finally, in this chapter we present results previously obtained on greatest common divisors and least common multiples of graphs. In Chapter II the concepts of prime graphs, prime divisors of graphs, and primeconnected graphs are presented. We show the existence of prime trees of any odd size and the existence of prime-connected trees that are not prime having any odd composite size. Then the number of prime divisors in a graph is studied. Finally, we present several results involving the existence of graphs whose size satisfies some prescribed condition and which contains a specified number of prime divisors. Chapter III presents properties of greatest common divisors and least common multiples of graphs. Then graphs with a prescribed number of greatest common divisors or least common multiples are studied. In Chapter IV we study the sizes of greatest common divisors and least common multiples of specified graphs. We find the sizes of greatest common divisors and least common multiples of stars and that of stripes. Then the size of greatest common divisors and least common multiples of paths and complete graphs are investigated. In particular, the size of least common multiples of paths versus K3 or K4 are determined. Then we present the greatest common divisor index of a graph and we determine this parameter for several classes of graphs. iii In Chapter V greatest common divisors and least common multiples of digraphs are introduced. The existence of least common mutliples of two stars is established, and the size of a least common multiple is found for several pairs of stars. Finally, we present the concept of greatest common divisor index of a digraph and determine it for several classes of digraphs. iv / Mathematical Sciences / Ph. D. (Mathematical sciences)

Στόχος της παρούσας διπλωματικής εργασίας είναι να παρουσιάσει τις τεχνικές και τις μεθόδους που πρέπει να ακολουθηθούν για να διαμεριστούν οι κορυφές ενός απλού γραφήματος σε ομάδες κορυφών, οι οποίες κατέχουν όμοιες ή παρόμοιες δομές σύνδεσης με άλλες ομάδες. Η διπλωματική εργασία αποτελείται από τέσσερα μέρη. Στο πρώτο μέρος της εργασίας δίνεται ο ορισμός του απλού γραφήματος, της ομαδοποίησης, του μπλοκ και του μοντέλου των μπλοκ, έννοιες οι οποίες είναι απαραίτητες για την συνέχεια. Στο δεύτερο μέρος περιγράφονται δύο τύποι ισοδυναμίας, που χρησιμοποιούνται για την ομαδοποίηση των κορυφών ενός γραφήματος, καθώς επίσης και δύο μέθοδοι για την επίτευξη της ομαδοποίησης των κορυφών. Στο τρίτο μέρος δίνεται μια πιο γενική ιδέα για την ομαδοποίηση των κορυφών ενός γραφήματος, η οποία βασίζεται σε μια από τις δυο προηγούμενες μεθόδους προσέγγισης της ομαδοποίησης. Στο τέταρτο και τελευταίο μέρος δίδεται μια επέκταση της ομαδοποίησης πάνω σε διμερή δεδομένα και παρουσιάζονται ατα αποτελέσματα, από την εφαρμογή αυτής της επέκτασης σε ένα εμπειρικό παράδειγμα. / -

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This thesis will study a variety of problems in graph theory. Initially, the focus will be on finding minimal degree conditions which guarantee the existence of various subgraphs. These subgraphs will all be formed of cycles, and this area of work will fall broadly into two main categories. First to be considered are cycles in edge-coloured graphs and, in particular, two questions of Li, Nikiforov and Schelp. It will be shown that a 2-edge-coloured graph with minimal degree at least 3n/4 either is isomorphic to the complete 4-partite graph with classes of order n/4, or contains monochromatic cycles of all lengths between 4 and n/2 (rounded up). This answers a conjecture of Li, Nikiforov and Schelp. Attention will then turn to the length of the longest monochromatic cycle in a 2-edge-coloured graph with minimal degree at least cn. In particular, a lower bound for this quantity will be proved which is asymptotically best possible. The next chapter of the thesis then shows that a hamiltonian graph with minimal degree at least (5-sqrt7)n/6 contains a 2-factor with two components. The thesis then concludes with a chapter about X_H, which is the number of copies of a graph H in the random graph G(n,p). In particular, it will be shown that, for a connected graph H, the value of X_H modulo k is approximately uniformly distributed, provided that k is not too large a function of n.

Στόχος της παρούσας διπλωματικής εργασίας είναι η μελέτη της ομαδοποίησης των προσημασμένων γράφων. / The main theme of this dissertation is the blockmodeling of signed graphs.

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A graph on at least k+1 vertices is said to have global connectivity k if any two of its vertices are connected by k independent paths. The local connectivity of two vertices is the number of independent paths between those specific vertices. This dissertation is concerned with pairwise connectivity notions, meaning that the focus is on local connectivity relations that are required for a number of or all pairs of vertices. We give a detailed overview about how uniformly k-connected and uniformly k-edge-connected graphs are related and provide a complete constructive characterization of uniformly 3-connected graphs, complementing classical characterizations by Tutte. Besides a tight bound on the number of vertices of degree three in uniformly 3-connected graphs, we give results on how the crossing number and treewidth behaves under the constructions at hand. The second central concern is to introduce and study cut sequences of graphs. Such a sequence is the multiset of edge weights of a corresponding Gomory-Hu tree. The main result in that context is a constructive scheme that allows to generate graphs with prescribed cut sequence if that sequence satisfies a shifted variant of the classical Erdős-Gallai inequalities. A complete characterization of realizable cut sequences remains open. The third central goal is to investigate the spectral properties of matrices whose entries represent a graph's local connectivities. We explore how the spectral parameters of these matrices are related to the structure of the corresponding graphs, prove bounds on eigenvalues and related energies, which are sums of absolute values of all eigenvalues, and determine the attaining graphs. Furthermore, we show how these results translate to ultrametric distance matrices and touch on a Laplace analogue for connectivity matrices and a related isoperimetric inequality.