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Paper   IPM / M / 15396
 School of Mathematics Title: The smarandache vertices of comaximal graph of a commutative ring Author(s): Amir Masoud rahimi (Joint with E. Mehdi-Nezhad) Status: To Appear Journal: Libertas Math. Supported by: IPM
Abstract:
Let R be a commutative ring with identity 1 Ì¸= 0. Define the comaximal graph of R, denoted by CG(R), to be the graph whose vertices are the elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. A vertex a in a simple graph G is said to be a Smarandache vertex (or S-vertex for short) provided that there exist three distinct vertices x, y, and b (all different from a) in G such that aâx, aâb, and bây are edges in G but there is no edge between x and y. The main object of this paper is to study the S-vertices of CG(R) and CG2(R)  J(R) (or CGJ (R) for short), where CG2(R) is the subgraph of CG(R) which consists of nonunit elements of R and J(R) is the Jacobson radical of R. There is also a discussion on a relationship between the diameter and S-vertices of CGJ (R).