A solution to Godsil's conjecture on the edge-connectivity of graphs in association schemes

TL;DR

This paper confirms Godsil's conjecture that the edge-connectivity of a connected regular equiarboreal graph in an association scheme equals its degree, using combinatorial and electrical network methods, and shows such graphs have perfect matchings if on an even number of vertices.

Contribution

It proves Godsil's conjecture and extends the result to all connected regular equiarboreal graphs, establishing their edge-connectivity equals their degree.

Findings

Edge-connectivity of connected regular equiarboreal graphs equals their degree.

Confirmed Godsil's conjecture on graphs in association schemes.

Connected regular equiarboreal graphs on even vertices have perfect matchings.

Abstract

A graph $G$ is called equiarboreal if the number of spanning trees containing a given edge in $G$ is independent of the choice of edge. In [Combinatorica 1(2) (1981) 163--167], Godsil proved that any graph which is a colour class in an association scheme is equiarboreal, and further conjectured that the edge-connectivity of a connected graph which is a colour class in an association scheme equals its vertex degree. In this paper, we confirm this long-standing conjecture. More generally, we prove an even stronger result that the edge-connectivity of a connected regular equiarboreal graph equals its degree by combinatorial and electrical network approaches. As a consequence, we show that every connected regular equiarboreal graph on an even number of vertices has a perfect matching.