Oriented spanning trees and stationary distribution of digraphs

TL;DR

This paper introduces reduction formulas for key properties of digraphs using biclique partitions, enabling efficient enumeration and extending existing results to more general digraph classes.

Contribution

It provides novel reduction formulas for oriented spanning trees, stationary distributions, and Kemeny's constant based on biclique partitions, broadening previous line digraph results.

Findings

Reduction formulas for oriented spanning trees and stationary distributions

Method for enumerating spanning trees using biclique partitions

Extension of line digraph results to general digraphs

Abstract

By using biclique partitions of digraphs, this paper gives reduction formulas for the number of oriented spanning trees, stationary distribution vector and Kemeny's constant of digraphs. As applications, we give a method for enumerating spanning trees of undirected graphs by vertex degrees and biclique partitions. The biclique partition formula also extends the results of Knuth and Levine from line digraphs to general digraphs.