Complement of the generalized total graph of commutative rings

Complement of the generalized total graph of commutative rings 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

Complement of the generalized total graph of commutative rings

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Forum D'Analystes, Chennai
Subject
Mathematics; Analysis; Functional Analysis; Abstract Harmonic Analysis; Special Functions; Fourier Analysis; Measure and Integration
ISSN
0971-3611
eISSN
2367-2501
D.O.I.
10.1007/s41478-018-0093-6
Publisher site
See Article on Publisher Site

Abstract

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

Journal

The Journal of AnalysisSpringer Journals

Published: Jun 4, 2018

References

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