Matroids from gain graphs over quotient groups

Abstract

We present a new construction for matroids from gain graphs that simultaneously generalizes several existing constructions. The construction takes as input a gain graph over a Frobenius group $\hat{I}“$ with Frobenius kernel $\hat{I}{“}_1$ and outputs an elementary lift of the frame matroid of the underlying gain graph over the quotient group $\hat{I}“/\hat{I}{“}_1$. While the hypothesis that $\hat{I}“$ is a Frobenius group may seem unusual, we prove that it is in some sense necessary: if $\hat{I}“$ is any finite group with a nontrivial proper normal subgroup $\hat{I}{“}_1$ and there is a construction that takes in a complete $\hat{I}“$-gain graph and outputs an elementary lift $M$ of the frame matroid of the underlying $(\hat{I}“/\hat{I}{“}_1)$-gain graph so that a cycle of the graph is a circuit of $M$ if and only if it is $\hat{I}“$-balanced, then $\hat{I}“$ is a Frobenius group with Frobenius kernel $\hat{I}{“}_1$.