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On the vertex connectivity of weakly zero-divisor graph of commutative rings

  • Mohd Shariq
  • Jitender Kumar
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Abstract

The weakly zero-divisor graph WΓ(R) of a commutative ring R is the simple undirected graph whose vertices are nonzero zero-divisors of R, and two distinct vertices x, y are adjacent if and only if there exists w∈ ann(x) and $ z\in {\rm ann}(y)$ such that wz =0. In this paper, first we prove that the vertex connectivity of WΓ(R) is equal to its minimum degree, where R is either an Artinian ring or reduced ring. For any finite ring R, we obtain the vertex connectivity of WΓ(R). Moreover, this paper characterizes all the vertices that attain the minimum degree of WΓ(R).

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Cite

BibTeXRIS
BibTeX
@misc{shariq2026vertex,
  title = {{On the vertex connectivity of weakly zero-divisor graph of commutative rings}},
  author = {Mohd Shariq and Jitender Kumar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.02103},
  eprint = {2609.02103},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02103},
}
RIS
TY  - UNPB
TI  - On the vertex connectivity of weakly zero-divisor graph of commutative rings
AU  - Mohd Shariq
AU  - Jitender Kumar
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.02103
UR  - https://arxiv.org/abs/2609.02103
AB  - The weakly zero-divisor graph WΓ(R) of a commutative ring R is the simple undirected graph whose vertices are nonzero zero-divisors of R, and two distinct vertices x, y are adjacent if and only if there exists w∈ ann(x) and $ z\in {\rm ann}(y)$ such that wz =0. In this paper, first we prove that the vertex connectivity of WΓ(R) is equal to its minimum degree, where R is either an Artinian ring or reduced ring. For any finite ring R, we obtain the vertex connectivity of WΓ(R). Moreover, this paper characterizes all the vertices that attain the minimum degree of WΓ(R).
ER  -