Vertex connectivity of the nonzero nonunit core of the comaximal graph of $\mathbb Z_n$

TL;DR

This paper determines the vertex connectivity of a specific induced subgraph of the comaximal graph of _n for squarefree n, showing it is maximally connected with explicit formulas and efficient algorithms.

Contribution

It provides an exact formula for the vertex connectivity of the nonzero nonunit core of the comaximal graph of _n when n is squarefree, using a novel representation and analysis.

Findings

_2 has vertex connectivity _{m-1}(p_i-1) for squarefree n

The graph is maximally connected and edge connectivity equals minimum degree

Distance, diameter, radius formulas, and a linear-time algorithm are established

Abstract

This article settles Problem 7.2 posed by [Banerjee, Special Matrices (2022)] for the induced subgraph $G_2$ of the comaximal graph $\Gamma(\mathbb Z_n)$ when $n$ is squarefree. Let $n=p_1p_2\cdots p_m$ with distinct primes $p_1<\cdots<p_m$, and let $G_2$ be the graph on the nonzero nonunit residue classes modulo $n$. We use Chinese remainder representation of $\mathbb Z_n$, and encodes each vertex by the set of vanishing coordinates. This converts $G_2$ into a weighted blow-up of a disjointness graph on nonempty proper subsets of $\{1,\dots,m\}$. Within this model, we derive exact class sizes, explicit degree formulas, the minimum-degree layer, and a short-path criterion. The main theorem proves the connectivity of $G_{2}$ as $\kappa(G_2)=\prod_{i=1}^{m-1}(p_i-1)=\tfrac{\phi(n)}{p_m-1}$. Consequently, earlier upper bound is sharp, $G_2$ is maximally connected, and its edge connectivity agrees with its minimum degree. We also obtain distance formulas, diameter and radius information, and a linear-time algorithm once the prime factorization is known.