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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 ̸= 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 a—x, a—b, and b—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).

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