Thesis (Ph. D.)--Rutgers University, 2008. / "Graduate Program in Mathematics." Includes bibliographical references (p. 82-83).

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. The concept is related to the energy of a class of molecules in chemistry and was first brought to mathematics by Gutman in 1978 ([8]). In this thesis, we do a comprehensive study on the energy of graphs and digraphs. In Chapter 3, we review some existing upper and lower bounds for the energy of a graph. We come up with some new results in this chapter. A graph with n vertices is hyper-energetic if its energy is greater than 2n−2. Some classes of graphs are proved to be hyper-energetic. We find a new class of hyper-energetic graphs which is introduced and proved to be hyper-energetic in Section 3.3. The energy of a digraph is the sum of the absolute values of the real part of the eigenvalues of its adjacency matrix. In Chapter 4, we study the energy of digraphs in a way that Pe˜na and Rada in [19] have defined. Some known upper and lower bounds for the energy of digraphs are reviewed. In Section 4.5, we bring examples of some classes of digraphs in which we find their energy. Keywords. Energy of a graph, hyper-energetic graph, energy of a digraph. / vii, 80 leaves ; 29 cm

Thesis (Ph.D.) -- University of Newcastle, 2003. / School of Mathematical and Physical Sciences. Includes bibliographical references (p. 161-162). "Also available online".

Thesis (M.Sc.) - Carleton University, 2005. / Includes bibliographical references (p. 61-63). Also available in electronic format on the Internet.

Thesis (M.S.)--Rochester Institute of Technology, 1994. / Typescript. Includes bibliographical references (leaves 77-80).

abstract: A tiling is a collection of vertex disjoint subgraphs called tiles. If the tiles are all isomorphic to a graph $H$ then the tiling is an $H$-tiling. If a graph $G$ has an $H$-tiling which covers all of the vertices of $G$ then the $H$-tiling is a perfect $H$-tiling or an $H$-factor. A goal of this study is to extend theorems on sufficient minimum degree conditions for perfect tilings in graphs to directed graphs. Corrádi and Hajnal proved that every graph $G$ on $3k$ vertices with minimum degree $delta(G)ge2k$ has a $K_3$-factor, where $K_s$ is the complete graph on $s$ vertices. The following theorem extends this result to directed graphs: If $D$ is a directed graph on $3k$ vertices with minimum total degree $delta(D)ge4k-1$ then $D$ can be partitioned into $k$ parts each of size $3$ so that all of parts contain a transitive triangle and $k-1$ of the parts also contain a cyclic triangle. The total degree of a vertex $v$ is the sum of $d^-(v)$ the in-degree and $d^+(v)$ the out-degree of $v$. Note that both orientations of $C_3$ are considered: the transitive triangle and the cyclic triangle. The theorem is best possible in that there are digraphs that meet the minimum degree requirement but have no cyclic triangle factor. The possibility of added a connectivity requirement to ensure a cycle triangle factor is also explored. Hajnal and Szemerédi proved that if $G$ is a graph on $sk$ vertices and $delta(G)ge(s-1)k$ then $G$ contains a $K_s$-factor. As a possible extension of this celebrated theorem to directed graphs it is proved that if $D$ is a directed graph on $sk$ vertices with $delta(D)ge2(s-1)k-1$ then $D$ contains $k$ disjoint transitive tournaments on $s$ vertices. We also discuss tiling directed graph with other tournaments. This study also explores minimum total degree conditions for perfect directed cycle tilings and sufficient semi-degree conditions for a directed graph to contain an anti-directed Hamilton cycle. The semi-degree of a vertex $v$ is $min{d^+(v), d^-(v)}$ and an anti-directed Hamilton cycle is a spanning cycle in which no pair of consecutive edges form a directed path. / Dissertation/Thesis / Ph.D. Mathematics 2013

Indiana University-Purdue University Indianapolis (IUPUI) / Subnetwork or path mining is an emerging data mining problem in many areas including scientific and commercial applications. Graph modeling is one of the effective ways in representing real world networks. Many natural and man-made systems are structured in the form of networks. Traditional machine learning and data mining approaches assume data as a collection of homogenous objects that are independent of each other whereas network data are potentially heterogeneous and interlinked. In this paper we propose a novel algorithm to find subnetworks and Maximal paths from a weighted, directed network represented as a graph. The main objective of this study is to find meaningful Maximal paths from a given network based on three key parameters: node weight, edge weight, and direction. This algorithm is an effective way to extract Maximal paths from a network modeled based on a user’s interest. Also, the proposed algorithm allows the user to incorporate weights to the nodes and edges of a biological network. The performance of the proposed technique was tested using a Colorectal Cancer biological network. The subnetworks and paths obtained through our network mining algorithm from the biological network were scored based on their biological significance. The subnetworks and Maximal paths derived were verified using MetacoreTM as well as literature. The algorithm is developed into a tool where the user can input the node list and the edge list. The tool can also find out the upstream and downstream of a given entity (genes/proteins etc.) from the derived Maximal paths. The complexity of finding the algorithm is found to be O(nlogn) in the best case and O(n^2 logn) in the worst case.

A new connection between combinatorics and noncommutative algebra is established by relating a certain class of directed graphs to noncommutative Koszul algebras. The directed graphs in this class are called full graphs and are defined by a set of criteria on the edges. The structural properties of full graphs are studied as they relate to the edge criteria. A method is introduced for generating a Koszul algebra Lambda from a full graph G. The properties of Lambda are examined as they relate to the structure of G, with special attention being given to the construction of a projective resolution of certain semisimple Lambda-modules based on the structural properties of G. The characteristics of the Koszul algebra Lambda that is derived from the product of two full graphs G' and G' are studied as they relate to the properties of the Koszul algebras Lambda' and Lambda' derived from G' and G'. / Ph. D.

This paper uncovers the relationship between stock markets and exchange rates in seven countries by employing stable aggregate currency (SAC) for the period of 1973- 2004. Ordinary Least Squares (OLS) regression, time series methods, and directed acyclic graphs are applied to the daily data on stock market indices and exchange rates. The findings based on regression analysis show that exchange rate exposure of stock markets is statistically significant when stock indexes in SAC are used. Using an innovation accounting technique, we confirm that stock markets and exchange rates are correlated. Moreover, in most cases stock markets are more exogenous than foreign currency markets, which explains the relatively high percentage of uncertainty in the foreign currency market. Overall, SAC-based models give relatively more accurate and robust results than those which employ stock indices in local currencies, because it is more accurate to convert both variables into the same denominator.

Thesis (B.A.)--Haverford College, Dept. of Computer Science, 2004. / Includes bibliographical references.