Browsing by Subject "Graph Theory"
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Item Even Harmonious Labelings of Disconnected Graphs(2015-05) Stewart, DanielleA graph $G$ with $q$ edges is called \textit{graceful} if there is an injection from the vertices of $G$ to the set $\{0,1,\ldots,q\}$ such that when each edge $xy$ is assigned the label $|f(x)-f(y)|$, the resulting edge labels are distinct. This notion as well as a number of other functions from a graph to a set of non-negative integers were studied as tools for decomposing the complete graph into isomorphic subgraphs. A graph $G$ with $q$ edges is said to be harmonious if there is an injection $f$ from the vertices of $G$ to the group of integers modulo $q$ such that when each edge $xy$ is assigned the label $f(x)+f(y)$ $($mod $q)$, the resulting edge labels are distinct. If $G$ is a tree, exactly one label may be used on two vertices. Over the years, many variations of these two concepts have been studied and hundreds of articles have been written on these topics. We study a variant of harmonious labeling. A function $f$ is said to be an even harmonious labeling of a graph $G$ with $q$ edges if $f$ is an injection from the vertices of $G$ to the integers from $0$ to $2q$ and the induced function $f^*$ from the edges of $G$ to ${0,2,\ldots,2(q-1)}$ defined by $f^*(xy)=f(x)+f(y) ($mod $2q)$ is bijective. Only a few papers have been written on even harmonious labeling. This paper focuses on finding even harmonious labelings for disjoint graphs. Among the families we investigate are: the disjoint union of cycles and stars, unions of cycles with paths, and unions of squares of paths.Item Gamma-Supermagic Labeling The Products of Two Cycles(2020-08) Sorensen, LincolnA $\Gamma$-supermagic group edge labeling of a graph $G(V,E)$ with $|E|=g$ is an injection from $E$ to an Abelian group of order $g$ such that the sum of labels of all incident edges of every vertex $v \in V$ is equal to the same element $\gamma \in \Gamma$. We develop methods for labeling the products of two cycles $C_m \square C_n$ using a variety of external direct products of Abelian groups.Item Graph Labelings and Tournament Scheduling(2015-05) Shepanik, AaronDuring my research I studied and became familiar with distance magic and distance antimagic labelings and their relation to tournament scheduling. Roughly speaking, the relation is as follows. Let the vertices on the graph represent teams in a tournament, and let an edge between two vertices a and b represent that team a will play team b in the tournament. Further, suppose we can rank the teams based on previous games, say, the preceding season. These integer rankings become labels for the vertices. Of particular interest were handicap tournaments, that is, tournaments designed to give each team a more balanced chance of winning.Item System Decomposition for Distributed Data-Driven Process Monitoring(2020-08) Khatib, ShaazEffective monitoring of chemical processes is required to ensure safe and economical operation. Data-driven monitoring methods are popular for detecting and diagnosing faults in chemical plants. An effective approach for implementing a data-driven monitoring method to detect and diagnose faults in large-scale systems such as chemical plants is to do so in a distributed configuration. A decomposition (i.e. an allocation of sensors among different sets called subsystems) must be selected before a monitoring method can be implemented in the distributed configuration. The monitoring method is then applied to each subsystem to come to a local decision as to whether a fault is detected or what the local diagnosis is. Finally, a consensus strategy uses the local decisions of the subsystems to reach a final fault decision. The performance of a fault detection method can be improved when it is implemented in the distributed configuration. Some of the operational constraints associated with implementing a monitoring method for a large-scale system can also be satisfied in the distributed configuration. For example, distributed methods can be implemented using computers installed at multiple locations if transmitting measurements from all the sensors to a single location is difficult. The ability of the distributed configuration to satisfy operational constraints and the performance of the distributed methods depend on the selected decomposition. The first objective is to propose methods which find a decomposition for which the performance of a distributed data-driven monitoring method in detecting or diagnosing a set of faults is near optimal subject to user-imposed constraints. The proposed methods use greedy search algorithms to generate many feasible candidate decompositions and subsystems and then use process data to directly evaluate the performance of these candidates to find a near optimal decomposition. The second objective is to propose methods which find the minimum number of locations required for distributed monitoring and the monitoring tasks that should be implemented at each location subject to user-imposed constraints. A commonality in the proposed methods is that they use algorithms from graph theory. The proposed methods are also fully automated and software for the methods is made available.