Semirings of graphs

Homomorphisms and applications in network problems

Authors

  • Saifur Rahman Rajiv Gandhi University.
  • Gete Umbrey Jawaharlal Nehru College.

DOI:

https://doi.org/10.22199/issn.0717-6279-5137

Keywords:

graph semirings, graph join, union and intersection, semigroup, homomorphisms

Abstract

This paper deals with studying some algebraic structures of the graphs as an attempt to visualize abstract mathematics. We have used some binary graph operations to investigated the algebraic structures of graphs with examples. This work emphasizes specifically the construction of semigroup or monoid and semiring, and their properties.

This manuscript also aims to give a focused introduction of a class of homomorphism on the semiring of graphs. Some instances of real-life decision problems are consequently discussed. This article is also in a nascent stage of relating number theory and graph theory through mappings.

Author Biographies

Saifur Rahman, Rajiv Gandhi University.

Department of Mathematics.

Gete Umbrey, Jawaharlal Nehru College.

Department of Mathematics.

References

M. Yasin Ali, “Some structures of Hemirings”, Pure and Applied Mathematics Journal, vol. 6, no. 1, pp. 45–50, 2017. https://doi.org/10.11648/j.pamj.20170601.16

R. El Bashir, J. Hurt, A. Jančařı́k, and T. Kepka, “Simple commutative semirings”, Journal of Algebra, vol. 236, no. 1, pp. 277–306, 2001. https://doi.org/10.1006/jabr.2000.8483

A. Bustamante, “Link Algebra: A New Approach to Graph Theory”, 2011. ArXiv: 1103.3539.

K. Glazek, A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences. Poland: Springer, 2002.

J. S. Golan, Semirings and their Applications. Netherlands: Springer, 1999.

J. S. Golan, Some Recent Applications of Semiring theory. International Conference on Algebra in Memory of Kostia Beider, National Cheng Kung University, Tainan, pp. 1-18, 2005.

M. Hazewinkel, Ed., Handbook of algebra, 6th ed. New York, NY: Elsevier, 1996.

U. Hebisch and H. J. Weinert, Semirings: Algebraic Theory and Applications in Computer Science. Series in Algebra, vol. 5. World Scientific, 1998. https://doi.org/10.1142/3903

Z. M. Kishka, M. Saleem, S. Sharqawy, and A. Elrawy, “On matrix of matrices over semirings”, Mathematical Sciences Letters, vol. 7, no. 2, pp. 107–110, 2018. https://doi.org/10.18576/msl/070205

Y.-F. Lin and J. S. Ratti, “Connectivity of the graphs of semirings: Lifting and product,”, Proceedings of the American Mathematical Society, vol. 24, no. 2, pp. 411–414, 1970. https://doi.org/10.1090/s0002-9939-1970-0274534-8

Y.-F. Lin and J. S. Ratti, “The graphs of Semirings”, Journal of Algebra, vol. 14, no. 1, pp. 73–82, 1970. https://doi.org/10.1016/0021-8693(70)90134-1

A. Mokhov and V. Khomenko, “Algebra of parameterised graphs”, ACM Transactions on Embedded Computing Systems, vol. 13, no. 4s, pp. 1–22, 2014. https://doi.org/10.1145/2627351

A. Mokhov, “Algebraic graphs with class (functional pearl)”, Proceedings of the 10th ACM SIGPLAN International Symposium on Haskell, 2017.https://doi.org/10.1145/3122955.3122956

D. Shurbert, An Introduction to Graph Homomorphisms. Department of Mathematics and Computer Science University of Puget Sound, 2013.

G. Umbrey and S. Rahman, “An Approach Towards Rank And Nullity Of Algebraic Expressions Of Graphs”, Journal of Advance Research in Dynamical and Control Systems, vol. 12 (Special Issue), pp. 35-45, 2020.

G. Umbrey and S. Rahman, “Application of Graph Semirings in Decision Networks”, Mathematical Forum, vol. 28, pp. 40-51, 2020.

G. Umbrey and S. Rahman, “Determining paths energy of a complex network”, Advances in Mathematics: Scientific Journal, vol. 9, no. 10, pp. 8761–8770, 2020. https://doi.org/10.37418/amsj.9.10.99

H. S. Vandiver, “Note on a simple type of algebra in which the cancellation law of addition does not hold”, Bulletin of the American Mathematical Society, vol. 40, no. 12, pp. 914–920, 1934. https://doi.org/10.1090/s0002-9904-1934-06003-8

Published

2022-10-28

How to Cite

[1]
S. Rahman and G. Umbrey, “Semirings of graphs: Homomorphisms and applications in network problems”, Proyecciones (Antofagasta, On line), vol. 41, no. 6, pp. 1273-1296, Oct. 2022.

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