The vertex connectivity of some classes of divisible design graphs

TL;DR

This paper investigates the vertex connectivity of certain divisible design graphs, revealing instances where the connectivity is less than the degree, including cases where it is less than any power of 2, highlighting new structural properties.

Contribution

The paper determines the vertex connectivity of specific classes of divisible design graphs and provides examples where connectivity is significantly lower than the degree.

Findings

Vertex connectivity can be less than the degree $k$.

Examples show connectivity less than any power of 2.

Identifies structural conditions affecting connectivity.

Abstract

A $k$-regular graph is called a divisible design graph if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbours, and two vertices from different classes have exactly $\lambda_2$ common neighbours. In this paper, we find the vertex connectivity of some classes of divisible design graphs, in particular, we present examples of divisible design graphs, whose vertex connectivity is less than $k$, where $k$ is the degree of a vertex. We also show that the vertex connectivity a divisible design graphs may be less than $k$ by any power of 2.