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.

翻译：本文解决了[Banerjee, Special Matrices (2022)]中关于$\mathbb Z_n$的共素图$\Gamma(\mathbb Z_n)$的诱导子图$G_2$在$n$无平方因子时的问题7.2。设$n=p_1p_2\cdots p_m$，其中$p_1<\cdots<p_m$为互异素数，$G_2$为模$n$的非零非单位剩余类构成的图。我们利用$\mathbb Z_n$的中国剩余表示，将每个顶点编码为消失坐标的集合。这便将$G_2$转化为一个关于$\{1,\dots,m\}$的非空真子集的不交图的加权膨胀。在此模型下，我们推导出精确的类大小、显式度公式、最小度层以及短路径准则。主要定理证明$G_{2}$的连通性满足$\kappa(G_2)=\prod_{i=1}^{m-1}(p_i-1)=\tfrac{\varphi(n)}{p_m-1}$。由此，先前给出的上界是紧的，$G_2$是最大连通的，且其边连通度等于最小度。我们还获得了距离公式、直径与半径信息，并在已知素数分解时给出一个线性时间算法。