Vertex and Edge Connectivity of the Zero Divisor graph \(\Gamma[\mathbb{Z}_n]\)

B. Surendranath Reddy; Rupali S. Jain; N. Laxmikanth

doi:10.26713/cma.v11i2.1319

Authors

B. Surendranath Reddy School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded, Maharashtra
Rupali S. Jain School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded, Maharashtra
N. Laxmikanth School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded, Maharashtra

DOI:

https://doi.org/10.26713/cma.v11i2.1319

Keywords:

Zero divisor graph, Vertex connectivity, Edge connectivity, Minimum degree

Abstract

The Zero divisor Graph \(\Gamma[R]\) of a commutative ring \(R\) is a graph with vertex set being the set of non-zero zero divisors of \(R\) and there is an edge between two vertices if their product is zero. In this paper, we prove that the vertex, edge connectivity and the minimum degree of the zero divisor graph \(\Gamma[\mathbb{Z}_n]\) for any natural number \(n\), are equal.

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References

M.R. Ahmadi and R. Jahani-Nezhad, Energy and Weiner index of zero-divisor graphs, Iranian Journal of Mathematical Chemistry 2(1) (2011), 45 – 51, DOI: 10.22052/ijmc.2011.5166.

S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative rings, Journal of Algebra 274 (2004), 847 – 855, DOI: 10.1016/S0021-8693(03)00435-6.

D. F. Anderson and P. S. Livingston, The zero-divisor graph of commutative ring, Journal of Algebra 217(2) (1999), 434 – 447, DOI: 10.1006/jabr.1998.7840.

I. Beck, Coloring of commutative rings, Journal of Algebra 116(1) (1988), 208 – 226, DOI: 10.1016/0021-8693(88)90202-5.

J. Clark and D. A. Holton, A First Look at Graph Theory, World Scientific (1991), DOI: 10.1142/1280.

B.S. Reddy, R.S. Jain and L. Nandala, Spectrum and Wiener index of the zero divisor graph (Gamma[mathbb{Z}n]), arXiv:1707.05083 [math.RA], URL https://arxiv.org/abs/1707.05083.