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Articles 1 - 16 of 16
Full-Text Articles in Mathematics
Problems In Chemical Graph Theory Related To The Merrifield-Simmons And Hosoya Topological Indices, William B. O'Reilly
Problems In Chemical Graph Theory Related To The Merrifield-Simmons And Hosoya Topological Indices, William B. O'Reilly
Electronic Theses and Dissertations
In some sense, chemical graph theory applies graph theory to various physical sciences. This interdisciplinary field has significant applications to structure property relationships, as well as mathematical modeling. In particular, we focus on two important indices widely used in chemical graph theory, the Merrifield-Simmons index and Hosoya index. The Merrifield-Simmons index and the Hosoya index are two well-known topological indices used in mathematical chemistry for characterizing specific properties of chemical compounds. Substantial research has been done on the two indices in terms of enumerative problems and extremal questions. In this thesis, we survey known extremal results and consider the generalized …
Gallai-Ramsey Number For Classes Of Brooms, Benjamin J. Hamlin
Gallai-Ramsey Number For Classes Of Brooms, Benjamin J. Hamlin
Electronic Theses and Dissertations
Given a graph $G$, we consider the problem of finding the minimum number $n$ such that any $k$ edge colored complete graph on $n$ vertices contains either a rainbow colored triangle or a monochromatic copy of the graph $G$, denoted $gr_k(K_{3}:G)$. More precisely we consider $G=B_{m,\ell}$ where $B_{m,\ell}$ is a broom graph with $m$ representing the number of vertices on the handle and $\ell$ representing the number of bristle vertices. We develop a technique to reduce the difficulty of finding $gr_{k}(K_{3}:B_{m,\ell})$, and use the technique to prove a few cases with a fixed handle length, but arbitrarily many bristles. Further, …
Conflict Free Connectivity And The Conflict-Free-Connection Number Of Graphs, Travis D. Wehmeier
Conflict Free Connectivity And The Conflict-Free-Connection Number Of Graphs, Travis D. Wehmeier
Electronic Theses and Dissertations
We explore a relatively new concept in edge-colored graphs called conflict-free connectivity. A conflict-free path is a (edge-) colored path that has an edge with a color that appears only once. Conflict-free connectivity is the maximal number of internally disjoint conflict-free paths between all pairs of vertices in a graph. We also define the c-conflict-free-connection of a graph G. This is the maximum conflict-free connectivity of G over all c-colorings of the edges of G. In this paper we will briefly survey the works related to conflict-free connectivity. In addition, we will use the probabilistic method to achieve a bound …
Taking A Canon To The Adjunction Formula, Paul M. Harrelson
Taking A Canon To The Adjunction Formula, Paul M. Harrelson
Electronic Theses and Dissertations
In this paper, we show how the canonical divisor of a graph is related to the canonical divisor of its subgraph. The use of chip firing and the adjunction formula for graphs ex- plains said relation and even completes it. We go on to show the difference between the formula for full subgraphs and that of non-full subgraphs. Examples are used to simplify these results and to see the adjunction formula in action. Finally, we show that though the adjunction formula seems simple at first glance, it is somewhat complex and rather useful.
Inverse Problems Related To The Wiener And Steiner-Wiener Indices, Matthew Gentry
Inverse Problems Related To The Wiener And Steiner-Wiener Indices, Matthew Gentry
Electronic Theses and Dissertations
In a graph, the generalized distance between multiple vertices is the minimum number of edges in a connected subgraph that contains these vertices. When we consider such distances between all subsets of $k$ vertices and take the sum, it is called the Steiner $k$-Wiener index and has important applications in Chemical Graph Theory. In this thesis we consider the inverse problems related to the Steiner Wiener index, i.e. for what positive integers is there a graph with Steiner Wiener index of that value?
Sparse Trees With A Given Degree Sequence, Ao Shen
Sparse Trees With A Given Degree Sequence, Ao Shen
Electronic Theses and Dissertations
In this thesis, we consider the properties of sparse trees and summarized a certain class of trees under some constraint (including with a given degree sequence, with given number of leaves, with given maximum degree, etc.) which have maximum Wiener index and the minimum number of subtrees at the same time. Wiener index is one of the most important topological indices in chemical graph theory. Steiner k�� Wiener index can be regarded as the generalization of Wiener index, when k = 2, Steiner Wiener index is the same as Wiener index. Steiner k�� Wiener index of a tree T is …
Dynamics Of Gene Networks In Cancer Research, Paul Scott
Dynamics Of Gene Networks In Cancer Research, Paul Scott
Electronic Theses and Dissertations
Cancer prevention treatments are being researched to see if an optimized treatment schedule would decrease the likelihood of a person being diagnosed with cancer. To do this we are looking at genes involved in the cell cycle and how they interact with one another. Through each gene expression during the life of a normal cell we get an understanding of the gene interactions and test these against those of a cancerous cell. First we construct a simplified network model of the normal gene network. Once we have this model we translate it into a transition matrix and force changes on …
Graph Invariants Of Trees With Given Degree Sequence, Rachel Bass
Graph Invariants Of Trees With Given Degree Sequence, Rachel Bass
Electronic Theses and Dissertations
Graph invariants are functions defined on the graph structures that stay the same under taking graph isomorphisms. Many such graph invariants, including some commonly used graph indices in Chemical Graph Theory, are defined on vertex degrees and distances between vertices. We explore generalizations of such graph indices and the corresponding extremal problems in trees. We will also briefly mention the applications of our results.
Pattern Containment In Circular Permutations, Charles Lanning
Pattern Containment In Circular Permutations, Charles Lanning
Electronic Theses and Dissertations
Pattern containment in permutations, as opposed to pattern avoidance, involves two aspects. The first is to contain every pattern at least once from a given set, known as finding superpatterns; while the second is to contain some given pattern as many times as possible, known as pattern packing. In this thesis, we explore these two questions in circular permutations and present some interesting observations. We also raise some questions and propose some directions for future study.
Combinatorics Of Compositions, Meghann M. Gibson
Combinatorics Of Compositions, Meghann M. Gibson
Electronic Theses and Dissertations
Integer compositions and related enumeration problems have been extensively studied. The cyclic analogues of such questions, however, have significantly fewer results. In this thesis, we follow the cyclic construction of Flajolet and Soria to obtain generating functions for cyclic compositions and n-color cyclic compositions with various restrictions. With these generating functions we present some statistics and asymptotic formulas for the number of compositions and parts in such compositions. Combinatorial explanations are also provided for many of the enumerative observations presented.
Gallai-Ramsey Number Of An 8-Cycle, Jonathan Gregory
Gallai-Ramsey Number Of An 8-Cycle, Jonathan Gregory
Electronic Theses and Dissertations
Given a graph G and a positive integer k, define the Gallai-Ramsey number to be the minimum number of vertices n such that any k-edge-coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we establish the Gallai-Ramsey number of an 8-cycle for all positive integers.
Combinatorial Optimization Of Subsequence Patterns In Words, Matthew R. Just
Combinatorial Optimization Of Subsequence Patterns In Words, Matthew R. Just
Electronic Theses and Dissertations
Packing patterns in words concerns finding a word with the maximum number of a prescribed pattern. The majority of the work done thus far is on packing patterns into permutations. In 2002, Albert, Atkinson, Handley, Holton and Stromquist showed that there always exists a layered permutation containing the maximum number of a layered pattern among all permutations of length n. Consequently, the packing density for all but two (up to equivalence) permutation patterns up to length 4 can be obtained. In this thesis we consider the analogous question for colored patterns and permutations. By introducing the concept of colored blocks …
Labeled Trees And Spanning Trees: Computational Discrete Mathematics And Applications, Demet Yalman
Labeled Trees And Spanning Trees: Computational Discrete Mathematics And Applications, Demet Yalman
Electronic Theses and Dissertations
In this thesis, we examine two topics. In the first part, we consider Leech tree which is a tree of order n with positive integer edge weights such that the weighted distances between pairs of vertices are exactly from 1 to n choose 2. Only five Leech trees are known and some non-existence results have been presented through the years. Variations of Leech trees such as the minimal distinct distance trees and modular Leech trees have been considered in recent years. In this thesis, such Leech-type questions on distances between leaves are studied as well as some other labeling questions …
Combinatorial Game Theory: An Introduction To Tree Topplers, John S. Ryals Jr.
Combinatorial Game Theory: An Introduction To Tree Topplers, John S. Ryals Jr.
Electronic Theses and Dissertations
The purpose of this thesis is to introduce a new game, Tree Topplers, into the field of Combinatorial Game Theory. Before covering the actual material, a brief background of Combinatorial Game Theory is presented, including how to assign advantage values to combinatorial games, as well as information on another, related game known as Domineering. Please note that this document contains color images so please keep that in mind when printing.
Graphs Of Classroom Networks, Rebecca Holliday
Graphs Of Classroom Networks, Rebecca Holliday
Electronic Theses and Dissertations
In this work, we use the Havel-Hakimi algorithm to visualize data collected from students to investigate classroom networks. The Havel-Hakimi algorithm uses a recursive method to create a simple graph from a graphical degree sequence. In this case, the degree sequence is a representation of the students in a classroom, and we use the number of peers with whom a student studied or collaborated to determine the degree of each. We expand upon the Havel-Hakimi algorithm by coding a program in MATLAB that generates random graphs with the same degree sequence. Then, we run another algorithm to find the isomorphism …
Precise Partitions Of Large Graphs, Pouria Salehi Nowbandegani
Precise Partitions Of Large Graphs, Pouria Salehi Nowbandegani
Electronic Theses and Dissertations
First by using an easy application of the Regularity Lemma, we extend some known results about cycles of many lengths to include a specified edge on the cycles. The results in this chapter will help us in rest of this thesis. In 2000, Enomoto and Ota posed a conjecture on the existence of path decomposition of graphs with fixed start vertices and fixed lengths. We prove this conjecture when |G| is large. Our proof uses the Regularity Lemma along with several extremal lemmas, concluding with an absorbing argument to retrieve misbehaving vertices. Furthermore, sharp minimum degree and degree sum conditions …