Inequivalent Representations Of 5-connected Non-graphic Frame Matroids

Matroid theory was introduced by Hassler Whitney in 1935.He aimed at axiomatizing the commonness of various concepts of dependence in linear algebra and graph theory.Because of the ingenious structure of matroids,almost all the results of graph theory without involving vertices and all the conclusions of linear independence not related to certain field can be expressed by matroids.Besides,matroid theory is closely related to topology,geometry,combinatorial optimization,coding theory and network theory.A matroid M is a frame matroid if there is a matroid M' with a basis B with M = M'\B such that every element of M is spanned by at most two elements of B.The class of frame matroids is minor-closed,which includes graphic matroids and plays an important role in matroid decomposition,extremal matroid theory,and the proof of Rota's Conjecture.Frame matroids were also introduced by Zaslavsky from the perspective of "biased graphs".We say that these biased graphs are their frame representation.A frame matroid may have many inequivalent frame representations.Chen,DeVos,Funk,and Pivotto considered all inequivalent frame representations of graphic frame matroids,and showed that some of them have exponentially many inequivalent frame representations.It is the real reason making to characterise the class of frame matroids very difficult.Chen and Geelen recently exhibit infinitely many non-5-connected excluded minors for the class of frame matroids(in arXiv:1703.04857,2017).However,Chen conjectured that there are only finitely many 7-connected excluded minors for the class of frame matroids.To prove this conjecture,we first must know all inequivalent frame representations of 5-connected non-graphic frame matroids,which is also the main aim of this thesis.To be specific,we prove that,if G and H are two inequivalent frame representations of a 5-connected non-graphic frame matroid,G and H have at most three vertices whose neighbours are not the same.This paper is organized as follows.In Chapter 1,some related definitions and our main results are introduced.The main result is proved in Chapter 2.In Chapter 3,some further research work is given.