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dc.contributor.advisorReff, Nathan
dc.contributor.authorCoble, Nolan J.
dc.date.accessioned2021-09-08T14:16:26Z
dc.date.available2021-09-08T14:16:26Z
dc.date.issued9/14/2020
dc.identifier.urihttp://hdl.handle.net/20.500.12648/6657
dc.description.abstractThe quaternions are a non-commutative division ring that extends the complex numbers. A gain graph is a simple graph together with a gain function that assigns a value from an arbitrary group to each edge of the graph. We can define certain concepts on these graphs such as adjacency and Laplacian matrices, gains of paths, and more. If we restrict ourselves to the unit norm quaternions, we can define quaternionic unit gain graphs, or U(H)-gain graphs, as gain graphs where the domain of the gain function is the unit quaternions. Traditional methods from spectral graph theory are not directly extended to quaternionic unit gain graphs due to non-commutativity. In this thesis, we extend a previous result from complex unit gain graphs, so that the right eigenvalues of the adjacency matrix for a U(H)-gain cycle can be written explicitly from the gain of the cycle. A thorough treatment of quaternions, quaternionic linear algebra, T-gain graphs, and U(H)-gain cycles is given. At the end, the right eigenvalues are calculated for a particular U(H)-gain cycle, and the results are compared to those obtained from a MATLAB method for approximating them.
dc.subjectGraph Theory
dc.subjectSpectral Graph Theory
dc.subjectQuaternions
dc.subjectLaplacian Eigenvalues
dc.titleSpectral Properties of Quaternionic Unit Gain Cycles
dc.typethesis
refterms.dateFOA2021-09-08T14:16:26Z
dc.description.institutionSUNY Brockport
dc.description.departmentMathematics
dc.source.statuspublished
dc.description.publicationtitleSenior Honors Theses
dc.contributor.organizationState University of New York College at Brockport
dc.languate.isoen_US


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