Let v ∈ Z+ and G be a simple graph. A G-decomposition of Kv is a collection F={F1,F2,...,Ft} of subgraphs of Kv such that every edge of Kv occurs in exactlyone of the subgraphs and every graph Fi ∈ F is isomorphic to G. A G-decomposition of Kv is called balanced if each vertex of Kv occurs in the same number of copies of G. In 2011, Dukes and Malloch provided an existence theory for balanced G-decompositions of Kv. Shortly afterwards, Bonisoli, Bonvicini, and Rinaldi introduced degree- and orbit-balanced G-decompositions. Similar to balanced decompositions,these two types of G-decompositions impose a local structure on the vertices of Kv. In this thesis, we will present an existence theory for degree- and orbit-balanced G-decompositions of Kv. To do this, we will first develop a theory for decomposing Kv into copies of G when G contains coloured loops. This will be followed by a brief discussion about the applications of such decompositions. Finally, we will explore anextension of this problem where Kv is decomposed into a family of graphs. We will examine the complications that arise with families of graphs and provide results for a few special cases. / Graduate

In this paper we show that specific classes of graphs are reconstructible; we explore the relationship between the. reconstruction and edge-reconstruction conjectures; we prove that several classes of graphs are actually Harary to the reconstructible; and we give counterexamples reconstruction and edge-reconstruction conjectures for infinite graphs.

Indiana University-Purdue University Indianapolis (IUPUI) / Research on social networks and understanding the interactions of the users can be modeled as a task of graph mining, such as predicting nodes and edges in networks. Dealing with such unstructured data in large social networks has been a challenge for researchers in several years. Neural Networks have recently proven very successful in performing predictions on number of speech, image, and text data and have become the de facto method when dealing with such data in a large volume. Graph NeuralNetworks, however, have only recently become mature enough to be used in real large-scale graph prediction tasks, and require proper structure and data modeling to be viable and successful. In this research, we provide a new modeling of the social network which captures the attributes of the nodes from various dimensions. We also introduce the Neural Network architecture that is required for optimally utilizing the new data structure. Finally, in order to provide a hot-start for our model, we initialize the weights of the neural network using a pre-trained graph embedding method. We have also developed a new graph embedding algorithm. We will first explain how previous graph embedding methods are not optimal for all types of graphs, and then provide a solution on how to combat those limitations and come up with a new graph embedding method.

Uncertain graph analysis plays a crucial role in many real-world applications, where the presence of uncertain information poses challenges for traditional graph mining algorithms. In this paper, we propose a novel method to find antagonistic communities in signed uncertain graphs, where vertices in the same community have a large expectation of positive edge weights and the vertices in different communities have a large expectation of negative edge weights. By restricting all the computations on small local parts of the signed uncertain graph, our method can efficiently find significant groups of antagonistic communities. We also provide theoretical foundations for the method. Extensive experiments on five real-world datasets and a synthetic dataset demonstrate the outstanding effectiveness and efficiency of the proposed method. / Thesis / Master of Science (MSc)

Both chordal and weakly chordal graphs have been topics of research in graph theory for many years. Upon reading their definitions it is clear that the weakly chordal class of graphs is a relaxation of the chordal condition for graphs. The question is then asked could we possibly find and study the properties if we, in turn, relaxed the weakly chordal condition for graphs? We start by providing the definitions and basic results needed later on. In the second chapter, we discuss perfect graphs, some of their properties, and some subclasses that were researched. The third chapter is focused on a new class of graphs, the definition of which relaxes the restrictions for chordal and weakly chordal graphs, and extends certain results from weakly chordal graphs to this class.

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A labeling of a graph G is said to be r-distinguishing if no automorphism of G preserves all of the vertex labels. The smallest such number r for which there is an r-distinguishing labeling on G is called the distinguishing number of G. The distinguishing set of a group Gamma, D(Gamma), is the set of distinguishing numbers of graphs G in which Aut(G) = Gamma. It is shown that D(Gamma) is non-empty for any finite group Gamma. In particular, D(D_n) is found where D_n is the dihedral group with 2n elements. From there, the generalized Petersen graphs, GP(n,k), are defined and the automorphism groups and distinguishing numbers of such graphs are given. / Master of Science

The expanding constant is a measure of graph connectivity that is important for certain applications. This paper discusses the mathematical foundations for the construction of Winnie Li's graphs and for the proof that Winnie Li's graphs are Ramanujan. The paper also establishes the implications of the Ramanujan property for the expanding constant. / Master of Science

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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)