Zero-sum flows of the linear lattice

A zero-sum flow of a graph G is an element of the nullspace of the incidence matrix of G whose entries are nonzero real numbers. A zero-sum flow is called a k-flow if all the entries of the nullspace vector are integers less than k in absolute value. It is conjectured that any graph with a zero-sum flow must admit a 6-flow. In this note, we consider the lattice of subspaces of an n  -dimensional vector space over a finite field. We prove the existence of zero-sum flows for the incidence matrix between two levels of the linear lattice with different rank numbers. Using field-theoretic considerations, we also show that there exists an ([m]q+1)([m]q+1)-flow or ([n−m]q+1[n−m]q+1)-flow between levels 1 and m   for 2≤m≤n−22≤m≤n−2 whenever m   or n−mn−m, respectively, divide n. Additionally, if neither m   nor n−mn−m divides n, we show there exists a 2- or 3-flow between levels 1 and m.