Young  reports that Murty  was the first to study matroids with all hyper-planes having the same size. Murty called such a matroid an "Equicardinal Matroid". Young renamed such a matroid a "Matroid Design". Further work on determining properties of these matroids was done by Edmonds, Murty, and Young [12, 20, 21]. These authors were able to connect the problem of determining the matroid designs with specified parameters with results on balanced incomplete block designs. The dual of a matroid design is one in which all circuits have the same size. Murty  restricted his attention to binary matroids and was able to characterize all connected binary matroids having circuits of a single size. Lemos, Reid, and Wu  provided partial information on the class of connected binary matroids having circuits of two different sizes. They also showed that there are many such matroids. In general, there are not many results that specify the matroids with circuits of just a few different sizes. Cordovil, Junior, and Lemos [2, 9] provided such results on matroids with small circumference. Here we determine the connected bicircular matroids with all circuits having the same size. We also provide structural information on the connected bicircular matroids with circuits of two different sizes. The bicircular matroids considered are in general non-binary. Hence these results are a start on extending Murty's characterization of binary matroid designs to non-binary matroids.
|Advisor:||Reid, Talmage J., Sheppardson, Laura|
|Commitee:||Labuda, Iwo, Wilkins, Dawn, Wu, Haidong|
|School:||The University of Mississippi|
|School Location:||United States -- Mississippi|
|Source:||DAI-B 72/04, Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Mathematics|
|Keywords:||Bicircular matroid, Circuits, Dual matroid design, Matroid design, Matroid dissertation, Single circuit spectrum|
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