J Anal https://doi.org/10.1007/s41478-018-0093-6 ORI G INAL RESEARCH PAPER Complement of the generalized total graph of commutative rings 1 1 Tamizh Chelvam Thirugnanam Balamurugan Mariappan Received: 28 January 2018 / Accepted: 18 May 2018 Forum D’Analystes, Chennai 2018 Abstract Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and H a nonempty proper multiplicative prime subset of R. The generalized total graph of R is the simple undirected graph GT ðRÞ with the vertex set R and two distinct vertices x and y are adjacent if and only if x þ y 2 H: In this paper, we investigate several graph theoretical properties of the complement GT ðRÞ: In particular, we obtain a characterization for GT ðRÞ to be claw-free or unicyclic or pancyclic. Also, we obtain the clique number and chromatic number of GT ðRÞ and discuss the perfect, planar and outer planarity nature for GT ðRÞ: Further, we dis- cuss various domination parameters for GT ðRÞ where P is a prime ideal of R. Keywords Commutative rings Total graph Complement Domination Gamma sets Mathematics Subject Classiﬁcation 05C75 05C25 13A15 13M05 1 Introduction Throughout this paper R denotes
The Journal of Analysis – Springer Journals
Published: Jun 4, 2018
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