Journal of Combinatorial Theory Series B ( IF 1.2 ) Pub Date : 2022-03-03 , DOI: 10.1016/j.jctb.2022.02.007 Daryl Funk 1 , Irene Pivotto 2 , Daniel Slilaty 3
Let M be a 3-connected matroid and let be a field. Let A be a matrix over representing M and let be a biased graph representing M. We characterize the relationship between A and , settling four conjectures of Zaslavsky. We show that for each matrix representation A and each biased graph representation of M, A is projectively equivalent to a canonical matrix representation arising from G as a gain graph over or realizing . Further, we show that the projective equivalence classes of matrix representations of M are in one-to-one correspondence with the switching equivalence classes of gain graphs arising from , except in one degenerate case.
中文翻译:
框架和提升图形拟阵的矩阵表示对应于增益函数
令M为 3 连通拟阵,令成为一个领域。设A是一个矩阵代表M并让是表示M的有偏图。我们描述了A和,解决了扎斯拉夫斯基的四个猜想。我们展示了对于每个矩阵表示A和每个有偏图表示的M , A投影等价于由G产生的规范矩阵表示,作为增益图要么意识到. 此外,我们证明了M的矩阵表示的投影等价类与增益图的切换等价类一一对应,除了一种退化的情况。