IJPAM: Volume 86, No. 6 (2013)
Department of Mathematics
Jadavpur University, Kolkata, 700032, INDIA
Abstract. For a ring , we define a simple undirected graph with all the non-zero elements of as vertices, and two vertices are adjacent if and only if either or or is a zero-divisor (including 0). We first consider its connectedness. Looking at , we determine the condition for connectedness of and also discuss its structure. We then consider connectedness, 2-connectedness and other properties of when is a direct product of rings. Giving particular attention to , we find out the degree patterns and consider girth, Eulerianity and planarity. Then we look at the non-commutative case of graph over the matrix rings and the infinite case of and , where is any ring .
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