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Abstract
Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by T
Γ(R), is the undirected graph with vertex set R and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). denotes the complement of T
Γ(R). The study on total graphs has been initiated by D. F. Anderson and A. Badawi [Citation2]. In this article, we characterize all commutative rings whose total graph (or its complement) is in some known class of graphs. Also we determine the structure
whenever |Reg(R)| = 2. Further, we obtain certain necessary conditions for
to be connected whenever
is connected and prove that
. It is also proved that if diam(T
Γ(R)) = 2, then T
Γ(R) is Hamiltonian, which is a generalization of a characterization proved by S. Akbari et al. [Citation1].
ACKNOWLEDGMENTS
The authors express their sincere thanks to the referee for exposing related studies in this direction and it has improved the presentation of the article. The work is supported by the INSPIRE programme(IF110072) of Department of Science and Technology, Government of India for the first author. Also the work reported here is supported by the UGC Major Research Project (F.No. 37-267/2009(SR)) awarded to the second author by the University Grants Commission, Government of India.
Notes
Communicated by I. Swanson.