2016/09/19 by Funk, Daryl, Pivotto, Irene, Slilaty, Daniel
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1609.05574
Let M be a 3-connected matroid and let \mathbb F be a field. Let A be a matrix over \mathbb F representing M and let (G,\mathcal B) be a biased graph representing M. We characterize the relationship between A and (G,\mathcal B), settling four conjectures of Zaslavsky. We show that for each matrix representation A and each biased graph representation (G,B) of M, A is projectively equivalent to a canonical matrix representation arising from G as a gain graph over \mathbb F+ or \mathbb F^× realizing B. 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 (G,\mathcal B), except in one degenerate case.