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Balanced Signed Total Graphs of Commutative Rings

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Abstract

This paper is primarily concerned with a natural extension of the notion of a total graph \(T(\varGamma (R))\) in the realm of signed graph for a finite commutative ring R. First, we characterize the rings for which the signed total graph \(T_{\varSigma }(\varGamma (R))\) and its line signed graph \(L(T_{\varSigma }(\varGamma (R)))\) are balanced. Second, we characterize the rings for which the negation of a signed total graph \(\eta (T_{\varSigma }(\varGamma (R)))\) is balanced. Several new directions for further research are also indicated.

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Acknowledgments

Authors would like to thank Professor Thomas Zaslavsky for his thought provoking suggestions and dedicate this paper to the sweet memories of Dr. B.D. Acharya. The first author is thankful to the Department of Atomic Energy (DAE) for providing the research grant vide sanctioned letter number: 2/39(26)/2012 -R&D-II/8861.

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Chhatra Marg, Delhi, 110007, India

    Pranjali Sharma

  2. Department of Mathematics, Kalasalingam University, Anand Nagar, KrishnanKoil, 626126, India

    Mukti Acharya

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  1. Pranjali Sharma
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  2. Mukti Acharya
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Correspondence to Pranjali Sharma.

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Sharma, P., Acharya, M. Balanced Signed Total Graphs of Commutative Rings. Graphs and Combinatorics 32, 1585–1597 (2016). https://doi.org/10.1007/s00373-015-1666-7

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  • DOI: https://doi.org/10.1007/s00373-015-1666-7

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