14 research outputs found
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 vertices is either or 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 ) 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 . 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 be a finite abelian group. Given , , any set of the form is called a {\em translate} of . A coloring of the elements of is {\em polychromatic} if every translate of contains an element of each color. The largest number of colors allowing an polychromatic coloring of the translates of is known as the {\em polychromatic number} of , denoted . 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 which does not contain a translate of ? 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 , denoted for all . 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 are presented. Results related to the polychromatic number of subsets of cardinality three and of and subsets of cardinality three of are also presented.</p
A polychromatic approach to a Turán-type problem in finite groups
Let be a finite abelian group. Given , , any set of the form is called a {\em translate} of . A coloring of the elements of is {\em polychromatic} if every translate of contains an element of each color. The largest number of colors allowing an polychromatic coloring of the translates of is known as the {\em polychromatic number} of , denoted . 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 which does not contain a translate of ? 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 , denoted for all . 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 are presented. Results related to the polychromatic number of subsets of cardinality three and of and subsets of cardinality three of are also presented