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Full-Text Articles in Discrete Mathematics and Combinatorics
Characteristic Sets Of Matroids, Dony Varghese
Characteristic Sets Of Matroids, Dony Varghese
Doctoral Dissertations
Matroids are combinatorial structures that generalize the properties of linear independence. But not all matroids have linear representations. Furthermore, the existence of linear representations depends on the characteristic of the fields, and the linear characteristic set is the set of characteristics of fields over which a matroid has a linear representation. The algebraic independence in a field extension also defines a matroid, and also depends on the characteristic of the fields. The algebraic characteristic set is defined in the similar way as the linear characteristic set.
The linear representations and characteristic sets are well studied. But the algebraic representations and …
Characterizations Of Certain Classes Of Graphs And Matroids, Jagdeep Singh
Characterizations Of Certain Classes Of Graphs And Matroids, Jagdeep Singh
LSU Doctoral Dissertations
``If a theorem about graphs can be expressed in terms of edges and cycles only, it probably exemplifies a more general theorem about matroids." Most of my work draws inspiration from this assertion, made by Tutte in 1979.
In 2004, Ehrenfeucht, Harju and Rozenberg proved that all graphs can be constructed from complete graphs via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. In Chapter 2, we consider the binary matroid analogue of each of these graph operations. We prove that the analogue of the result of Ehrenfeucht et. al. does …