On the zero forcing number of the complement of graphs with forbidden subgraphs

Motivated in part by an observation that the zero forcing number for the complement of a tree on $n$ vertices is either $n-3$ or $n-1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under some conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs.Comment: 17 pages, 8 figure

Using Markov chains to determine expected propagation time for probabilistic zero forcing

Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this studied. Given a Markov transition matrix for a probabilistic zero forcing process, we establish an exact formula for expected propagation time. We apply Markov chains to determine bounds on expected propagation time for various families of graphs

The Strong Spectral Property of Graphs: Graph Operations and Barbell Partitions

The utility of a matrix satisfying the Strong Spectral Property has been well established particularly in connection with the inverse eigenvalue problem for graphs. More recently the class of graphs in which all associated symmetric matrices possess the Strong Spectral Property (denoted $G^{SSP}$) were studied, and along these lines we aim to study properties of graphs that exhibit a so-called barbell partition. Such a partition is a known impediment to membership in the class $G^{SSP}$. In particular we consider the existence of barbell partitions under various standard and useful graph operations

Families of graphs with maximum nullity equal to zero forcing number

The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i =6 j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartite 3- semiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to find bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen graphs

A polychromatic approach to a Turán-type problem in finite groups

Let $G$ be a finite abelian group. Given $S \subseteq G$, $a \in G$, any set of the form $a + S = \{a \} + S$ is called a {\em translate} of $S$. A coloring of the elements of $G$ is {\em $S-$polychromatic} if every translate of $S$ contains an element of each color. The largest number of colors allowing an $S-$polychromatic coloring of the translates of $S$ is known as the {\em polychromatic number} of $S$, denoted $p_G(S)$. Determining the polychromatic number of finite abelian groups is a relatively new and unexplored method that can be used to solve the following problem: What is the maximum number of elements in a subset of $G$ which does not contain a translate of $S$? This type of problem called a Tur\'{a}n-type problem is common in extremal graph theory, but is new to the realm of algebra. This dissertation aims to determine bounds on the desired maximum number of elements, referred to as the {\em Tur\'{a}n number}, using polychromatic colorings on the desired maximum number of elements within the context of the well known abelian group the integers modulo $n$, denoted $\mathbb{Z}_n$ for all $n \geq 3$. The problem is also redefined and explored within the context of nonabelian groups such as the dihedral group and the dicyclic group.\\ \indent Trivial bounds are first presented on the Tur\'{a}n number for any group. Results on the improvement of the trivial lower bound are then presented. The results involve determining the polychromatic number for various subsets. The polychromatic number of any subset of cardinality two of any group is determined. Results related to the polychromatic number of any subset of odd prime cardinality of $\mathbb{Z}_n$ are presented. Results related to the polychromatic number of subsets of cardinality three and $n$ of $D_{2n}$ and subsets of cardinality three of $Dic_n$ are also presented.</p

A polychromatic approach to a Turán-type problem in finite groups

Let $G$ be a finite abelian group. Given $S \subseteq G$, $a \in G$, any set of the form $a + S = \{a \} + S$ is called a {\em translate} of $S$. A coloring of the elements of $G$ is {\em $S-$polychromatic} if every translate of $S$ contains an element of each color. The largest number of colors allowing an $S-$polychromatic coloring of the translates of $S$ is known as the {\em polychromatic number} of $S$, denoted $p_G(S)$. Determining the polychromatic number of finite abelian groups is a relatively new and unexplored method that can be used to solve the following problem: What is the maximum number of elements in a subset of $G$ which does not contain a translate of $S$? This type of problem called a Tur\\u27{a}n-type problem is common in extremal graph theory, but is new to the realm of algebra. This dissertation aims to determine bounds on the desired maximum number of elements, referred to as the {\em Tur\\u27{a}n number}, using polychromatic colorings on the desired maximum number of elements within the context of the well known abelian group the integers modulo $n$, denoted $\mathbb{Z}_n$ for all $n \geq 3$. The problem is also redefined and explored within the context of nonabelian groups such as the dihedral group and the dicyclic group.\\ \indent Trivial bounds are first presented on the Tur\\u27{a}n number for any group. Results on the improvement of the trivial lower bound are then presented. The results involve determining the polychromatic number for various subsets. The polychromatic number of any subset of cardinality two of any group is determined. Results related to the polychromatic number of any subset of odd prime cardinality of $\mathbb{Z}_n$ are presented. Results related to the polychromatic number of subsets of cardinality three and $n$ of $D_{2n}$ and subsets of cardinality three of $Dic_n$ are also presented