Peg solitaire is a game in which pegs are placed in every hole but one and the player jumps over pegs along rows or columns to remove them. Usually, the goal is to have a single peg remaining. In a 2011 paper, this game is generalized to graphs. In this paper, we consider a variation in which each peg must be jumped twice in order to be removed. For this variation, we consider the solvability of several graph families. For our major results, we characterize solvable joins of graphs and show that the Cartesian product of solvable graphs is likewise solvable.

Truss decomposition is an essential problem in graph mining, which focuses on discovering dense subgraphs of a graph. Detecting trusses in deterministic graphs is extensively studied in the literature. As most of the real-world graphs, such as social, biological, and communication networks, are associated with uncertainty, it is of great importance to study truss decomposition in a probabilistic context. However, the problem has received much less attention in a probabilistic framework. Furthermore, due to computational challenges of truss decomposition in probabilistic graphs, state-of- the-art approaches are not scalable to large graphs. Formally, given a user-defined threshold k (for truss denseness), we are interested in finding all the maximal subgraphs, which are a k-truss with high probability. In this thesis, we introduce a novel approach based on an asynchronous h-index updating process, which offers significant improvement over the state-of-the-art. Our extensive experimental results confirm the scalability and efficiency of our approach. / Graduate

New techniques to measure the information contained within a network of interconnected nodes (such as links between computers in the Internet) have recently been developed. This work studies the relationship between the computer time needed to solve a common network problem and the information contained within the given network.

This thesis introduces a novel application of factor graphs to the domain of analog circuits. It proposes a technique of leveraging factor graphs for performing statistical yield analysis of analog circuits that is much faster than the standard Monte Carlo/Simulation Program With Integrated Circuit Emphasis (SPICE) simulation techniques. We have designed a tool chain to model an analog circuit and its corresponding factor graph and then use a Gaussian message passing approach along the edges of the graph for yield calculation. The tool is also capable of estimating unknown parameters of the circuit given known output statistics through backward message propagation in the factor graph. The tool builds upon the concept of domain-specific modeling leveraged for modeling and interpreting different kinds of analog circuits. Generic Modeling Environment (GME) is used to design modeling environment for analog circuits. It is a configurable tool set that supports creation of domain-specific design environments for different applications. This research has developed a generalized methodology that could be applied towards design automation of different kinds of analog circuits, both linear and nonlinear. The tool has been successfully used to model linear amplifier circuits and a nonlinear Metal Oxide Semiconductor Field Effect Transistor (MOSFET) circuit. The results obtained by Monte Carlo simulations performed on these circuits are used as a reference in the project to compare against the tool's results. The tool is tested for its efficiency in terms of time and accuracy against the standard results.

During the past few decades, examinations of social, biological, and communication networks have taken on increased attention. While numerous models of these networks have arisen, some have lacked visual representations. This is particularly true in ecology, where scientists have often been restricted to at most three dimensions when creating graphical representations of pattern and process. I will introduce an application of dot product representation graphs that allows scientists to view the high dimensional connections in ecological networks. Using actual data, example graphs will be developed and analyzed using key measures of graph theory.

The results of a round robin tournament can be represented as a matrix of zeros and ones, by ordering the players and placing a one in the (i,j) position if player i beat player j, and zeros otherwise. These matrices, called tournament matrices, can be represented by graphs, called tournament graphs. They have been the subject of much research and study, yet there have been few attempts to give a wide exposition on the subject. Those that have been done tend to focus on the graph theoretical aspects of tournaments. S. Ree and Y. Koh did write a brief survey from the matrix viewpoint in 1998, but it was not complete and not published. This paper is an attempt to give an exposition on tournament matrices. Recent research will be presented, some new ideas and properties will be proposed, and a few applications of the material will be reviewed.

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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.

For a graph G, a set S is a dominating set if every vertex in V-S has a neighbor in S. A vertex contained in some minimum dominating set is called good; otherwise it is bad. A graph G has g(G) good vertices and b(G) bad vertices. The relationship between the order of G and g(G) assigns the graph to one of four classes. Our results include a method of classifying caterpillars. Further, we develop realizability conditions for a graph G given a triple of nonnegative integers representing the domination number of γ(G), g(G), and b(G), respectively, and provide constructions of graphs meeting those conditions. We define the goodness index of a vertex v in a graph G as the ratio of distinct γ(G)-sets containing v to the total number of γ(G)-sets, and provide formulas that yield the goodness index of any vertex in a given path.

This thesis is separated into two parts: threshold graphs and inertia sets. First we present an algorithmic approach to finding the minimum rank of threshold graphs and then progress to counting the number of threshold graphs with a specific minimum rank. Second, we find an algorithmic and more automated way of determining the inertia set of graphs with seven or fewer vertices using theorems and lemmata found in previous papers. Inertia sets are a relaxation of the inverse eigenvalue problem. Instead of determining all the possible eigenvalues that can be obtained by matrices with a specific zero/nonzero pattern we restrict to counting the number of positive and negative eigenvalues.