International Journal of Pure and Applied Mathematics Volume 86 No. 6 2013, 893-904 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v86i6.2 P A ijpam.eu THE GRAPH Γ 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 Γ 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 Z n , we determine the condition for connectedness of Γ 2 (Z n ) and also discuss its structure. We then consider connectedness, 2-connectedness and other properties of Γ 2 (R) when R is a direct product of rings. Giving particular attention to Γ 2 (Z n ), we find out the degree patterns and consider girth, Eulerianity and planarity. Then we look at the non-commutative case of Γ 2 graph over the matrix rings and the infinite case of Γ 2 (Z) and Γ 2 (Z × R), where R is any ring = {0}. AMS Subject Classification: 05C25 Key Words: ring, zero-divisor, zero-divisor graph, total graph, connected, complete, girth, diameter 1. Introduction Among graphs associated to a ring, Zero-divisor graph holds a special place due to the fact that it has revealed that the set of Zero-divisors of a ring, which algebraically does not possess a nice structure, has significant and interesting graph-theoretic structures. After the concept of zero-divisor graph was intro- duced by Beck [3], it was defined by D.F. Anderson and Livingston [2] in the following way : Let R be a commutative ring with 1. Then the zero-divisor graph of R, denoted as Γ(R), is the simple undirected graph with all the non- Received: May 9, 2013 c  2013 Academic Publications, Ltd. url: www.acadpubl.eu