The relationship between Kirchhoff graphs and equitable edge partitions of their corresponding digraphs is discussed. It is shown that if the natural edge partition for the associated digraph to a vector graph is equitable and the quotient matrix based on this partition is symmetric, then the vector graph is Kirchhoff. The converse is not true: many Kirchhoff graphs have natural edge partitions that are not equitable. In addition, it is shown that for a digraph with an equitable edge partition, the partition is uniform if and only if the quotient matrix is symmetric. Hence every uniform equitable edge partition of a digraph is a Kirchhoff partition and can generate a Kirchhoff graph.(c) 2022 Elsevier Inc. All rights reserved.
NSTL
Elsevier
