🤖 AI Summary
This study investigates the vertex connectivity of the induced subgraph $G_2$, comprising nonzero non-units, in the comaximal graph of the ring $\mathbb{Z}_n$ where $n$ is square-free. By leveraging the Chinese Remainder Theorem to represent elements as coordinate vectors, $G_2$ is modeled as a weighted expansion of a set-disjointness graph. The authors precisely determine the vertex connectivity of $G_2$ to be $\prod_{i=1}^{m-1}(p_i - 1)$, thereby establishing that $G_2$ achieves maximal connectivity—equal to its minimum degree—and derive key structural properties including distance, diameter, and radius. Furthermore, they present a linear-time algorithm based on the prime factorization of $n$, which confirms the tightness of previously known upper bounds and affirms the optimal connectivity of this class of graphs.
📝 Abstract
This article settles Problem 7.2 posed by [Banerjee, Special Matrices (2022)] for the induced subgraph $G_2$ of the comaximal graph $Γ(\mathbb Z_n)$ 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 $κ(G_2)=\prod_{i=1}^{m-1}(p_i-1)=\tfrac{φ(n)}{p_m-1}$. 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.