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

TL;DR

The paper resolves Godsil's conjecture by proving that every connected $k$-regular equiarboreal graph has edge-connectivity $\lambda(G)=k$, using a synthesis of combinatorial edge-cut analysis and electrical-network techniques anchored in resistance distances. The approach combines Foster's formula, the principle of substitution, and complete-bipartite to double-star transformations to bound $\Omega_G(u,v)$ and exclude smaller edge-cuts across progressively larger degrees. The authors systematically verify the claim for $k=3,4,5,6,7$ and then generalize to all $k\ge8$, introducing notions like strongly $S_{x,y}$-free edge cuts to control possible configurations. As a corollary, any connected regular equiarboreal graph on an even number of vertices contains a perfect matching, and the results reinforce the broader link between equiarboreal graphs, association schemes, and graph connectivity in algebraic combinatorics.

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.

Figures (8)

• Figure 1: The equivalent double star $S^{\omega}_{m,n}$.
• Figure 2: (a) The structure example of $G[C]$, (b) The weighted graph $K^{\omega}_4$ equivalent to $G^*$.
• Figure 3: The corresponding graphs $G[C]$ for edge cuts $C$ with $|C|=3$.
• Figure 4: The corresponding graphs $G[C]$ for edge cuts $C$ with $|C|=4$.
• Figure 5: (a) The network $M_1$, (b) The network $M_2$, (c) The network $M_3$.

Theorems & Definitions (49)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4