IJPAM: Volume 86, No. 6 (2013)

THE GRAPH $\Gamma_2(R)$ OVER A RING $R$

Raibatak Sen Gupta
Department of Mathematics
Jadavpur University, Kolkata, 700032, INDIA


Abstract. For a ring $R$, we define a simple undirected graph $\Gamma_2(R)$ with all the non-zero elements of $R$ as vertices, and two vertices $a, b$ are adjacent if and only if either $ab=0$ or $ba=0$ or $a+b$ is a zero-divisor (including 0). We first consider its connectedness. Looking at $\mathbb Z_n$, we determine the condition for connectedness of $\Gamma_2(\mathbb Z_n)$ and also discuss its structure. We then consider connectedness, 2-connectedness and other properties of $\Gamma_2(R)$ when $R$ is a direct product of rings. Giving particular attention to $\Gamma_2(\mathbb Z_n)$, we find out the degree patterns and consider girth, Eulerianity and planarity. Then we look at the non-commutative case of $\Gamma_2$ graph over the matrix rings and the infinite case of $\Gamma_2(\mathbb Z)$ and $\Gamma_2(\mathbb Z \times R)$, where $R$ is any ring $\not=\{0\}$.

Received: May 9, 2013

AMS Subject Classification: 05C25

Key Words and Phrases: ring, zero-divisor, zero-divisor graph, total graph, connected, complete, girth, diameter

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DOI: 10.12732/ijpam.v86i6.2 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 86
Issue: 6