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

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

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.


Keywords


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

Full Text:

PDF

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.




DOI: http://dx.doi.org/10.26713%2Fcma.v11i2.1319

Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905