For a graph G=(V,E), a set S⊆V is a global supply set if every vertex v∈V\S has at least one neighbor, say u, in S such that u has at least as many neighbors in S as v has in V \S. The global supply number is the minimum cardinality of a global supply set, denoted γgs (G). We introduce global supply sets and determine the global supply number for selected families of graphs. Also, we give bounds on the global supply number for general graphs, trees, and grid graphs.

A (closed) neighborhood-restricted 2-achromatic-coloring of a graph G is an assignment of colors to the vertices of G such that no more than two colors are assigned in any closed neighborhood. In other words, for every vertex v in G, the vertex v and its neighbors are in at most two different color classes. The 2-achromatic number is defined as the maximum number of colors in any 2-achromatic-coloring of G. We study the 2-achromatic number. In particular, we improve a known upper bound and characterize the extremal graphs for some other known bounds.

The structure of a protein molecule is highly correlated to its function. Some diseases such as cystic fibrosis are the result of a change in the structure of a protein so that this change interferes or inhibits its function. Often these changes in structure are caused by a misfolding of the protein molecule. To assist computational biologists, there is a database of proteins together with their misfolded versions, called decoys, that can be used to test the accuracy of protein structure prediction algorithms. In our work we use a nested graph model to quantify a selected set of proteins that have two single misfold decoys. The graph theoretic model used is a three tiered nested graph. Measures based on the vertex weights are calculated and we compare the quantification of the proteins with their decoys. Our method is able to separate the misfolded proteins from the correctly folded proteins.

abstract: Autism spectrum disorder (ASD) is a developmental neuropsychiatric condition with early childhood onset, thus most research has focused on characterizing brain function in young individuals. Little is understood about brain function differences in middle age and older adults with ASD, despite evidence of persistent and worsening cognitive symptoms. Functional Magnetic Resonance Imaging (MRI) in younger persons with ASD demonstrate that large-scale brain networks containing the prefrontal cortex are affected. A novel, threshold-selection-free graph theory metric is proposed as a more robust and sensitive method for tracking brain aging in ASD and is compared against five well-accepted graph theoretical analysis methods in older men with ASD and matched neurotypical (NT) participants. Participants were 27 men with ASD (52 +/- 8.4 years) and 21 NT men (49.7 +/- 6.5 years). Resting-state functional MRI (rs-fMRI) scans were collected for six minutes (repetition time=3s) with eyes closed. Data was preprocessed in SPM12, and Data Processing Assistant for Resting-State fMRI (DPARSF) was used to extract 116 regions-of-interest defined by the automated anatomical labeling (AAL) atlas. AAL regions were separated into six large-scale brain networks. This proposed metric is the slope of a monotonically decreasing convergence function (Integrated Persistent Feature, IPF; Slope of the IPF, SIP). Results were analyzed in SPSS using ANCOVA, with IQ as a covariate. A reduced SIP was in older men with ASD, compared to NT men, in the Default Mode Network [F(1,47)=6.48; p=0.02; 2=0.13] and Executive Network [F(1,47)=4.40; p=0.04; 2=0.09], a trend in the Fronto-Parietal Network [F(1,47)=3.36; p=0.07; 2=0.07]. There were no differences in the non-prefrontal networks (Sensory motor network, auditory network, and medial visual network). The only other graph theory metric to reach significance was network diameter in the Default Mode Network [F(1,47)=4.31; p=0.04; 2=0.09]; however, the effect size for the SIP was stronger. Modularity, Betti number, characteristic path length, and eigenvalue centrality were all non-significant. These results provide empirical evidence of decreased functional network integration in pre-frontal networks of older adults with ASD and propose a useful biomarker for tracking prognosis of aging adults with ASD to enable more informed treatment, support, and care methods for this growing population. / Dissertation/Thesis / Masters Thesis Biomedical Engineering 2019

A fundamental issue in many distributed computing problems is the need for nodes to distinguish themselves from their neighbors in a process referred to as symmetry breaking. Many well-known problems such as Maximal Independent Set (MIS), t-Ruling Set, Maximal Matching, and (\Delta+1)-Coloring, belong to the class of problems that require symmetry breaking. These problems have been studied extensively in the LOCAL model, which assumes arbitrarily large message sizes, but not as much in the CONGEST and k-machine models, which assume messages of size O(log n) bits. This dissertation focuses on finding upper and lower bounds for symmetry breaking problems, such as MIS and t-Ruling Set, in these congested models. Chapter 2 shows that an MIS can be computed in O(sqrt{log n loglog n}) rounds for graphs with constant arboricity in the CONGEST model. Chapter 3 shows that the t-ruling set problem, for t \geq 3, can be computed in o(log n) rounds in the CONGEST model. Moreover, it is shown that a 2-ruling set can be computed in o(log n) rounds for a large range of values of the maximum degree in the graph. In the k-machine model, k machines must work together to solve a problem on an arbitrary n-node graph, where n is typically much larger than k. Chapter 4 shows that any algorithm in the BEEP model (which assumes 'primitive' single bit messages) with message complexity M and round complexity T can be simulated in O(t(M/k^2 + T) poly(log n)) rounds in the k-machine model. Using this result, it is shown that MIS, Minimum Dominating Set (MDS), and Minimum Connected Dominating Set (MCDS) can all be solved in O(poly(log n) m/k^2) rounds in the k-machine model, where 'm' is the number of edges in the input graph. It is shown that a 2-ruling set can be computed even faster, in O((n/k^2+ k) poly(log n)) rounds, in the k-machine model. On the other hand, using information theoretic techniques and a reduction to a communication complexity problem, an \Omega(n/(k^2 poly(log n))) rounds lower bound for MIS in the k-machine model is also shown. As far as we know, this is the first example of a lower bound in the k-machine model for a symmetry breaking problem. Chapter 5 focuses on the Max Clique problem in the CONGEST model. Max Clique is trivially solvable in one round in the LOCAL model since each node can share its entire neighborhood with all neighbors in a single round. However, in the CONGEST model, nodes have to choose what to communicate and along what communication links. Thus, in a sense, they have to break symmetry and this is forced upon them by the bandwidth constraints. Chapter 5 shows that an O(n^{3/5})-approximation to Max Clique in the CONGEST model can be computed in O(1) rounds. This dissertation ends with open questions in Chapter 6.

An isohedral tiling is a tiling of congruent polygons that are also transitive, which is to say the configuration of degrees of vertices around each face is identical. Regular tessellations, or tilings of congruent regular polygons, are a special case of isohedral tilings. Viewing these tilings as graphs in planes, both Euclidean and non-Euclidean, it is possible to pose various problems of enumeration on the respective graphs. In this paper, we investigate some near-regular isohedral tilings of triangles and quadrilaterals in the hyperbolic plane. For these tilings we enumerate vertices as classified by number of edges in the shortest path to a given origin, by combinatorially deriving their respective generating functions.

We examine the problem of counting interval graphs. We answer the question posed by Hanlon, of whether the formal power series generating function of the number of interval graphs on n vertices has a positive radius of convergence. We have found that it is zero. We have obtained a lower bound and an upper bound on the number of interval graphs on n vertices. We also study the application of interval graphs to the dynamic storage allocation problem. Dynamic storage allocation has been shown to be NP-complete by Stockmeyer. Coloring interval graphs on-line has applications to dynamic storage allocation. The most colors used by Kierstead's algorithm is 3 ω -2, where ω is the size of the largest clique in the graph. We determine a lower bound on the colors used. One such lower bound is 2 ω -1.

What is the cohomology of a graph? Cohomology is a topological invariant and encodes such information as genus and euler characteristic. Graphs are combinatorial objects which may not a priori admit a natural and isomorphism invariant cohomology ring. In this project, given any finite graph G, we constructively define a cohomology ring H*(G) of G. Our method uses graph associahedra and toric varieties. Given a graph, there is a canonically associated convex polytope, called the graph associahedron, constructed from G. In turn, a convex polytope uniquely determines a toric variety. We synthesize these results, and describe the cohomology of the associated variety directly in terms of the graph G itself.

A path on n vertices, denoted by Pn, is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the order. A fourth power path, Pn4, is obtained from Pn by adding edges between any two vertices, u and v, whose distance in Pn, denoted by dPn(u,v), is less than or equal to four. The diameter of a graph G, denoted diam(G) is the greatest distance between any two distinct vertices of G. A radio labeling of a graph G is a function f that assigns to each vertex a label from the set {0,1,2,...} such that |f(u)−f(v)| ≥ diam(G)−d(u,v)+1 holds for any two distinct vertices, u and v in G (i.e., u, v ∈ V (G)). The greatest value assigned to a vertex by f is called the span of the radio labeling f, i.e., spanf =max{f(v) : v ∈ V (G)}. The radio number of G, rn(G), is the minimum span of f over all radio labelings f of G. In this paper, we provide a lower bound for the radio number of the fourth power path.

A k-majority tournament is a directed graph that models a k-majority voting scenario, which is realized by 2k - 1 rankings, called linear orderings, of the vertices in the tournament. Every k-majority voting scenario can be modeled by a tournament, but not every tournament is a model for a k-majority voting scenario. In this thesis we show that all acyclic tournaments can be realized as 2-majority tournaments. Further, we develop methods to realize certain quadratic residue tournaments as k-majority tournaments. Thus, each tournament within these classes of tournaments is a model for a k-majority voting scenario. We also explore important structures specifically pertaining to 2- and 3-majority tournaments and introduce the idea of pseudo-3-majority tournaments and inherited 2-majority tournaments.