Physical Sciences and Mathematics Commons^™

Random Walks On Digraphs, J.J.P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Let V = {1, · · · n} be a vertex set and S a non-negative row-stochastic matrix (i.e. rows sum to 1). V and S define a digraph G = G(V, S) and a directed graph Laplacian L as follows. If (S)ij > 0 (in what follows we will leave out the parentheses) there is a directed edge j → i. Thus the ith row of S identifies the edges coming into vertex i and their weights. This set of vertices are collectively the neighbors of i, and is denoted by Ni . The diagonal elements Sii are chosen such …

Behavior Of Petrie Lines In Certain Edge-Transitive Graphs, Ruby A. Chick

Math Theses

We survey the construction and classification of one-, two- and infinitely-ended members of a class of highly symmetric, highly connected infinite graphs. In addition, we pose a conjecture concerning the relationship between the Petrie lines and ends of some infinitely-ended members of this class.

Merging Peg Solitaire In Graphs, John Engbers, Ryan Weber

Mathematics, Statistics and Computer Science Faculty Research and Publications

Peg solitaire has recently been generalized to graphs. Here, pegs start on all but one of the vertices in a graph. A move takes pegs on adjacent vertices x and y, with y also adjacent to a hole on vertex z, and jumps the peg on x over the peg ony to z, removing the peg on y. The goal of the game is to reduce the number of pegs to one.

We introduce the game merging peg solitaire on graphs, where a move takes pegs on vertices x and z (with a hole on y) and merges them to …

Network Modeling Of Infectious Disease: Transmission, Control And Prevention, Christina M. Chandler

Honors College Theses

Many factors come into play when it comes to the transmission of infectious diseases. In disease control and prevention, it is inevitable to consider the general population and the relationships between individuals as a whole, which calls for advanced mathematical modeling approaches.

We will use the concept of network flow and the modified Ford-Fulkerson algorithm to demonstrate the transmission of infectious diseases over a given period of time. Through our model one can observe what possible measures should be taken or improved upon in the case of an epidemic. We identify key nodes and edges in the resulted network, which …

On T-Restricted Optimal Rubbling Of Graphs, Kyle Murphy

Electronic Theses and Dissertations

For a graph G = (V;E), a pebble distribution is defined as a mapping of the vertex set in to the integers, where each vertex begins with f(v) pebbles. A pebbling move takes two pebbles from some vertex adjacent to v and places one pebble on v. A rubbling move takes one pebble from each of two vertices that are adjacent to v and places one pebble on v. A vertex x is reachable under a pebbling distribution f if there exists some sequence of rubbling and pebbling moves that places a pebble on x. A pebbling distribution where every …

Average Shortest Path Length In A Novel Small-World Network, Andrea J. Allen

Honors Papers

We study a novel model of random graph which exhibits the structural characteristics of the Watts- Strogatz small-world network. The small-world network is characterized by a high level of local clustering while also having a relatively small graph diameter. The same behavior that makes the Watts-Strogatz model behave like this also makes it difficult to analyze. Our model addresses this issue, closely mimicking the same structure experimentally while following a constructive process that makes it easier to analyze mathematically. We present a bound on the average shortest path length in our new model, which we approach by looking at the …

Tying The Knot: Applications Of Topology To Chemistry, Tarini S. Hardikar

Honors Theses

Chirality (or handedness) is the property that a structure is “different” from its mirror image. Topology can be used to provide a rigorous framework for the notion of chirality. This project examines various types of chirality and discusses tools to detect chirality in graphs and knots. Notable theorems that are discussed in this work include ones that identify chirality using properties of link polynomials (HOMFLY polynomials), rigid vertex graphs, and knot linking numbers. Various other issues of chirality are explored, and some specially unique structures are discussed. This paper is borne out of reading Dr. Erica Flapan’s book, When Topology …

Network Analytics For The Mirna Regulome And Mirna-Disease Interactions, Joseph Jayakar Nalluri

Theses and Dissertations

miRNAs are non-coding RNAs of approx. 22 nucleotides in length that inhibit gene expression at the post-transcriptional level. By virtue of this gene regulation mechanism, miRNAs play a critical role in several biological processes and patho-physiological conditions, including cancers. miRNA behavior is a result of a multi-level complex interaction network involving miRNA-mRNA, TF-miRNA-gene, and miRNA-chemical interactions; hence the precise patterns through which a miRNA regulates a certain disease(s) are still elusive. Herein, I have developed an integrative genomics methods/pipeline to (i) build a miRNA regulomics and data analytics repository, (ii) create/model these interactions into networks and use optimization techniques, motif …

Complex Valued Graphs For Soft Computing, Florentin Smarandache, W.B. Vasantha Kandasamy, Ilanthenral K

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce in a systematic way the notion of complex valued graphs, strong complex valued graphs and complex neutrosophic valued graphs. Several interesting properties are defined, described and developed. Most of the conjectures which are open in case of usual graphs continue to be open problems in case of both complex valued graphs and strong complex valued graphs. We also give some applications of them in soft computing and social networks. At this juncture it is pertinent to keep on record that Dr. Tohru Nitta was the pioneer to use complex valued graphs …