Vertex connectivity of the nonzero nonunit core of the comaximal graph of $\mathbb Z_n$
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913083732000768 |
|---|---|
| author | Rather, Bilal Ahmad |
| author_facet | Rather, Bilal Ahmad |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01565 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vertex connectivity of the nonzero nonunit core of the comaximal graph of $\mathbb Z_n$ Rather, Bilal Ahmad Combinatorics Discrete Mathematics Group Theory Primary 05C40, Secondary 05C75, 13A99, 05C69 F.2.2 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. |
| title | Vertex connectivity of the nonzero nonunit core of the comaximal graph of $\mathbb Z_n$ |
| topic | Combinatorics Discrete Mathematics Group Theory Primary 05C40, Secondary 05C75, 13A99, 05C69 F.2.2 |
| url | https://arxiv.org/abs/2605.01565 |