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dc.contributor.authorLevit, Maxwell 12:53:19 (GMT) 12:53:19 (GMT)
dc.description.abstractThis thesis addresses the extent of the covering graph construction. How much must a cover X resemble the graph Y that it covers? How much can X deviate from Y? The main statistics of X and Y which we will measure are their regularity, the spectra of their adjacency matrices, and the length of their shortest cycles. These statistics are highly interdependent and the main contribution of this thesis is to advance our understanding of this interdependence. We will see theorems that characterize the regularity of certain covering graphs in terms of the number of distinct eigenvalues of their adjacency matrices. We will see old examples of covers whose lack of short cycles is equivalent to the concentration of their spectra on few points, and new examples that indicate certain limits to this equivalence in a more general setting. We will see connections to many combinatorial objects such as regular maps, symmetric and divisible designs, equiangular lines, distance-regular graphs, perfect codes, and more. Our main tools will come from algebraic graph theory and representation theory. Additional motivation will come from topological graph theory, finite geometry, and algebraic topology.en
dc.publisherUniversity of Waterlooen
dc.subjectalgebraic graph theoryen
dc.subjectcovering graphsen
dc.subjectdistance-regular graphsen
dc.titleGraph Coverings with Few Eigenvalues or No Short Cyclesen
dc.typeDoctoral Thesisen
dc.pendingfalse and Optimizationen and Optimizationen of Waterlooen
uws-etd.degreeDoctor of Philosophyen
uws.contributor.advisorChris, Godsil
uws.contributor.affiliation1Faculty of Mathematicsen

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