Many individuals with chronic post-stroke aphasia respond favorably to language therapy. However, treatment outcomes are highly variable and the neural mechanisms that support recovery, and perhaps explain this variability, remain elusive. Neuroimaging studies involving patients with aphasia have implicated a number of cortical regions in both hemispheres in post-treatment language processing, which suggests that network approaches may reveal important insights into the neural bases of language recovery. This dissertation investigated how functional activation, functional connectivity, and graph theoretical measures of network topology relate to treatment-induced changes in language functions. In the first study, we used an fMRI picture-naming task to examine functional activation in 26 patients with aphasia before and after 12 weeks of naming treatment. Changes in activation were associated with treatment outcomes, such that activation increased in patients who responded best to treatment (i.e., responders) but remained largely unchanged in patients who responded less favorably (i.e., nonresponders). In the second study, we analyzed functional connectivity and graph properties of an expanded picture naming network. Relative to healthy controls, patients had reduced functional connectivity, particularly within the left hemisphere and between regions in the left and right hemisphere. As in study 1, we found differential patterns of connectivity depending on treatment outcomes, such that connectivity normalized (i.e., became more like that of healthy controls) in responders but remained abnormally low in nonresponders. Similar results were obtained via the graph analysis. Finally, in the third study, we aimed to determine if pre-treatment global and local properties reflecting integration and segregation in a task-based semantic processing network predicted patients’ response to treatment. Network strength and global efficiency were significant predictors of improvement. Additionally, responders and nonresponders showed significant differences in nodal properties in a subset of bilaterally distributed regions in the frontal and parietal lobes. The results of these studies indicate that there are critical regional and network-level differences between patients who respond well to treatment and those who respond poorly, and that some of these differences can be identified before treatment is initiated. These results provide a foundation for further investigation of network-related biomarkers for recovery and updated models of recovery that account for pre-existing differences in network topology. / 2021-12-31T00:00:00Z

Given a finite set V, a convexity, C, is a collection of subsets of V that contains both the empty set and the set V and is closed under intersections. The elements of C are called convex sets. We can define several different convexities on the vertex set of a graph. In particular, the digital convexity, originally proposed as a tool for processing digital images, is defined as follows: a subset S of V(G) is digitally convex if, for every vertex v in V(G), we have N[v] a subset of N[S] implies v in S. Or, in other words, each vertex v that is not in the digitally convex set S needs to have a private neighbour in the graph with respect to S. In this thesis, we focus on the generation and enumeration of digitally convex sets in several classes of graphs. We establish upper bounds on the number of digitally convex sets of 2-trees, k-trees and simple clique 2-trees, as well as conjecturing a lower bound on the number of digitally convex sets of 2-trees and a generalization to k-trees. For other classes of graphs, including powers of cycles and paths, and Cartesian products of complete graphs and of paths, we enumerate the digitally convex sets using recurrence relations. Finally, we enumerate the digitally convex sets of block graphs in terms of the number of blocks in the graph, rather than in terms of the order of the graph. / Graduate

In this paper, we give necessary and sufficient conditions for the existence of a decomposition of the λ-fold mixed complete graph into partial orientations of P4 and S3. Simple direct constructions are given in each case.

Let v ∈ Z+ and G be a simple graph. A G-decomposition of Kv is a collection F={F1,F2,...,Ft} of subgraphs of Kv such that every edge of Kv occurs in exactlyone of the subgraphs and every graph Fi ∈ F is isomorphic to G. A G-decomposition of Kv is called balanced if each vertex of Kv occurs in the same number of copies of G. In 2011, Dukes and Malloch provided an existence theory for balanced G-decompositions of Kv. Shortly afterwards, Bonisoli, Bonvicini, and Rinaldi introduced degree- and orbit-balanced G-decompositions. Similar to balanced decompositions,these two types of G-decompositions impose a local structure on the vertices of Kv. In this thesis, we will present an existence theory for degree- and orbit-balanced G-decompositions of Kv. To do this, we will first develop a theory for decomposing Kv into copies of G when G contains coloured loops. This will be followed by a brief discussion about the applications of such decompositions. Finally, we will explore anextension of this problem where Kv is decomposed into a family of graphs. We will examine the complications that arise with families of graphs and provide results for a few special cases. / Graduate

In this paper we show that specific classes of graphs are reconstructible; we explore the relationship between the. reconstruction and edge-reconstruction conjectures; we prove that several classes of graphs are actually Harary to the reconstructible; and we give counterexamples reconstruction and edge-reconstruction conjectures for infinite graphs.

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Given a finite simple graph G, one can associate to G an ideal I_G, called the toric ideal of G. There are a number of algebraic invariants of ideals which are frequently studied in commutative algebra. In general, understanding these invariants is very difficult for arbitrary ideals. However, when the ideals are related to combinatorial objects, in this case, graphs, a deeper investigation can be conducted. If, in addition, the graph G is bipartite, even more can be said about these invariants. In this thesis, we explore a comparison of invariants of toric ideals of bipartite graphs. Our main result describes all possible values for the tuple (reg(K[E]/I_G), deg(h_{K[E]/I_G}), pdim(K[E]/I_G), depth(K[E]/I_G), dim(K[E]/I_G)) when G is a bipartite graph on n ≥ 1 vertices. / Thesis / Master of Science (MSc)

Firefighter is a discrete-time dynamic process that models the spread of a virus or rumour through a network. The name “Firefighter” arises from the initial analogy being the spread of fire among the vertices of a graph. Given a graph G, the process begins at time t = 0 when one or more vertices of G spontaneously “catch fire”. At each subsequent time step, a collection of b ≥ 1 “firefighters” defend a set of vertices which are not burning, and then the fire spreads from each burning vertex to all of its undefended neighbours. There are many possible objectives one could have, for example minimizing the expected number of vertices burned when the fire breaks out at a random location or locations, finding the maximum number of vertices that can be saved from burning if the fire breaks out at known locations, minimizing the length of the process, or bounding the proportion of vertices that can be saved from burning. It is also possible to consider multiple objectives that may be in conflict. There are a great number of papers in the literature which address these, and other, issues in terms of computational complexity, algorithms, approximation, asymptotics, heuristics, and more. The main purpose of this thesis is to survey developments on Firefighter and its variants which have appeared in the literature subsequent to a previous survey that appeared in 2009 [S. Finbow and G. MacGillivray. The firefighter problem: A survey of results, directions and questions. Australas. J. Comb., 43, 2009]. The thesis concludes with a list of open problems and future directions from the previous survey, annotated with references for papers that have made progress on those topics since then. / Graduate

A set D of vertices in a graph G is a 2-dominating set of G if every vertex in V − D has at least two neighbors in D. The 2-domination number of a graph G, denoted by γ2(G), is the minimum cardinality of a 2- dominating set of G. In this thesis, we discuss the 2-domination number of a special family of trees, called caterpillars. A caterpillar is a graph denoted by Pk(x1, x2, ..., xk), where xi is the number of leaves attached to the ith vertex of the path Pk. First, we present the 2-domination number of some classes of caterpillars. Second, we consider several types of complete caterpillars. Finally, we consider classification of caterpillars with respect to their spine length and 2-domination number.

Both chordal and weakly chordal graphs have been topics of research in graph theory for many years. Upon reading their definitions it is clear that the weakly chordal class of graphs is a relaxation of the chordal condition for graphs. The question is then asked could we possibly find and study the properties if we, in turn, relaxed the weakly chordal condition for graphs? We start by providing the definitions and basic results needed later on. In the second chapter, we discuss perfect graphs, some of their properties, and some subclasses that were researched. The third chapter is focused on a new class of graphs, the definition of which relaxes the restrictions for chordal and weakly chordal graphs, and extends certain results from weakly chordal graphs to this class.