A VARIATION OF ZERO-DIVISOR GRAPHS
Journal Title: Discussiones Mathematicae - General Algebra and Applications - Year 2015, Vol 35, Issue 2
Abstract
In this paper, we define a new graph for a ring with unity by extending the definition of the usual ‘zero-divisor graph’. For a ring R with unity, Γ1(R) is defined to be the simple undirected graph having all non-zero elements of R as its vertices and two distinct vertices x, y are adjacent if and only if either xy = 0 or yx = 0 or x + y is a unit. We consider the conditions of connectedness and show that for a finite commutative ring R with unity, Γ1(R) is connected if and only if R is not isomorphic to Z3 or Zk2 (for any k ∈ N − {1}). Then we characterize the rings R for which Γ1(R) realizes some well-known classes of graphs, viz., complete graphs, star graphs, paths (i.e., Pn), or cycles (i.e., Cn). We then look at different graph-theoretical properties of the graph Γ1(F), where F is a finite field. We also find all possible Γ1(R) graphs with at most 6 vertices.
Authors and Affiliations
Raibatak Sen Gupta, M. K. Sen, Shamik Ghosh
Keywords
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