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We discuss the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, formal power series, idealization of the R-module M and relations between the total graph of the ring R and its extensions are also dealt with.
Let R be a commutative ring with identity and )) ( ( R T Γ its total graph. The subject of this article is the investigation of the properties of the corresponding line graph ))). ( ( ( R T L Γ In particular, we determine the girth and clique number of ))). ( ( ( R T L Γ In addition to that, we find the condition for ))) ( ( ( R T L Γ to be Eulerian.
A generalization of the total graph of a ring is presented. Various properties are proved and some relations to the total graph of a ring as well as to the zero-divisor graph are established.