Physical Sciences and Mathematics Commons^™

Δ-Small Intersection Graphs Of Modules, Ahmed H. Alwan

Al-Bahir Journal for Engineering and Pure Sciences

Let R be a commutative ring with unit and M be a unitary left R-module. The δ-small intersection graph of non-trivial submodules of , denoted by , is an undirected simple graph whose vertices are the non-trivial submodules of , and two vertices are adjacent if and only if their intersection is a -small submodule of . In this article, we study the interplay between the algebraic properties of , and the graph properties of such as connectivity, completeness and planarity. Moreover, we determine the exact values of the diameter and girth of , as well as give a formula …

Ramsey Numbers For Connected 2-Colorings Of Complete Graphs, Mark Budden

Theory and Applications of Graphs

In 1978, David Sumner introduced a variation of Ramsey numbers by restricting to 2-colorings in which the subgraphs spanned by edges in each color are connected. This paper continues the study of connected Ramsey numbers, including the evaluation of several cases of trees versus complete graphs.

Matroid Generalizations Of Some Graph Results, Cameron Crenshaw

LSU Doctoral Dissertations

The edges of a graph have natural cyclic orderings. We investigate the matroids for which a similar cyclic ordering of the circuits is possible. A full characterization of the non-binary matroids with this property is given. Evidence of the difficulty of this problem for binary matroids is presented, along with a partial result for binary orderable matroids.

For a graph G, the ratio of |E(G)| to the minimum degree of G has a natural lower bound. For a matroid M that is representable over a finite field, we generalize this to a lower bound on …

Connectivity Of Matroids And Polymatroids, Zachary R. Gershkoff

LSU Doctoral Dissertations

This dissertation is a collection of work on matroid and polymatroid connectivity. Connectivity is a useful property of matroids that allows a matroid to be decomposed naturally into its connected components, which are like blocks in a graph. The Cunningham-Edmonds tree decomposition further gives a way to decompose matroids into 3-connected minors. Much of the research below concerns alternate senses in which matroids and polymatroids can be connected. After a brief introduction to matroid theory in Chapter 1, the main results of this dissertation are given in Chapters 2 and 3. Tutte proved that, for an element e of a …

Graph-Theoretic Simplicial Complexes, Hajos-Type Constructions, And K-Matchings, Julianne Vega

Theses and Dissertations--Mathematics

A graph property is monotone if it is closed under the removal of edges and vertices. Given a graph G and a monotone graph property P, one can associate to the pair (G,P) a simplicial complex, which serves as a way to encode graph properties within faces of a topological space. We study these graph-theoretic simplicial complexes using combinatorial and topological approaches as a way to inform our understanding of the graphs and their properties.

In this dissertation, we study two families of simplicial complexes: (1) neighborhood complexes and (2) k-matching complexes. A neighborhood complex is a simplicial …

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 …

Chromatic Connectivity Of Graphs, Elliot Laforge

Dissertations

No abstract provided.

Properly Colored Notions Of Connectivity - A Dynamic Survey, Xueliang Li, Colton Magnant

Theory and Applications of Graphs

A path in an edge-colored graph is properly colored if no two consecutive edges receive the same color. In this survey, we gather results concerning notions of graph connectivity involving properly colored paths.