Arborescences of Random Covering Graphs

TL;DR

This paper derives a formula for the expected weighted sum of arborescences in a random covering graph, advancing understanding of their structure and confirming a conjecture by Chepuri et al.

Contribution

It provides a closed-form expression for the expected arborescences in random covering graphs, linking them to the original graph's arborescences and resolving a prior conjecture.

Findings

Derived a formula for the expected weighted sum of arborescences

Confirmed the conjecture of Chepuri et al. regarding covering graphs

Established a connection between arborescences of covering graphs and original graphs

Abstract

A rooted arborescence of a directed graph is a spanning tree directed towards a particular vertex. A recent work of Chepuri et al. showed that the arborescences of a covering graph of a directed graph G are closely related to the arborescences of G. In this paper, we study the weighted sum of arborescences of a random covering graph and give a formula for the expected value, resolving a conjecture of Chepuri et al.