LINEAR DIOPHANTINE GRAPHS

Amr Nasr; Mohamed Anwar; Mohammed Abd-El Azim Seoud; Ahmed Abd Elgawad Elawady Elsonbaty

LINEAR DIOPHANTINE GRAPHS

Abstract

{This manuscript introduces linear Diophantine labelling, a new method for assigning labels to the vertices of finite, simple, undirected graphs. A key feature of this method is a divisibility condition imposed on the edges, incorporating number-theoretic properties into graph labelling. The study focuses on identifying maximal graphs that admit such labellings and computes their number of edges and the degree of each vertex. Number-theoretic techniques are employed to examine structural properties, including the characterization of maximum degree vertices and conditions for nonadjacent vertices. The manuscript also establishes necessary and sufficient conditions for vertices with equal degrees, offering new insights into the interaction between graph theory and number theory.

Keywords

Thanks

The author would like to extend their gratitude to the referees for their valuable feedback, insightful comments, and constructive suggestions, which significantly contributed to improving the quality and clarity of this work.

References

[1] Bachman, G., (1964), Introduction to p-Adic Numbers and Valuation Theory, Academic Press, New York.
[2] Bickle, A., (2020), Fundamentals of Graph Theory, American Mathematical Society, Providence, Rhode Island, USA.
[3] Burton, D. M., (2011), Elementary Number Theory, 7th ed., The McGraw-Hill Companies, New York.
[4] Catherine, A. and Palani, K., (2017), Indegree Prime Labelling of Digraphs, International Journal of Science, Engineering and Management, 12(12), pp. 358-361.
[5] Deretsky, T., Lee, S. M. and Mitchem, J., (1991), On vertex prime labellings of graphs in Graph Theory, Combinatorics and Applications, Alavi, J., Chartrand, G., Oellerman, O. and Schwenk, A., eds., Proceedings 6th International Conference Theory and Applications of Graphs, Wiley, New York, 1, pp. 359-369.
[6] Gallian, J. A., (2024), A Dynamic Survey of Graph Labelling, 27th ed., The Electronic Journal of Combinatorics, Duluth, Minnesota, U.S.A..
[7] Harary, F., (1969), Graph Theory, Addison-Wesley Publishing Company, Reading, Massachusetts.
[8] Pandya, P. and Shrimali, N. P., (2018), Vertex-edge neighborhood prime labelling of some graphs, International Journal of Science Research and Review, 7(10), pp. 735-743.

Details

Primary Language

English

Subjects

Algebra and Number Theory, Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)

Journal Section

Research Article

Authors

Amr Nasr ^* This is me
0009-0005-6857-1832
Egypt

Mohamed Anwar
0000-0003-0668-4071
Egypt

Mohammed Abd-El Azim Seoud This is me
0000-0003-3216-5873
Egypt

Ahmed Abd Elgawad Elawady Elsonbaty This is me
0000-0002-9001-4961
Egypt

Publication Date

June 9, 2026

Submission Date

April 16, 2025

Acceptance Date

August 12, 2025

Published in Issue

Year 2026 Volume: 16 Number: 6

IZ

https://izlik.org/JA78RZ84ZA

Cite

RIS / Bibtex

APA

Nasr, A., Anwar, M., Seoud, M. A.-E. A., & Elsonbaty, A. A. E. E. (2026). LINEAR DIOPHANTINE GRAPHS. TWMS Journal of Applied and Engineering Mathematics, 16(6), 771-785. https://izlik.org/JA78RZ84ZA

AMA

1.Nasr A, Anwar M, Seoud MAEA, Elsonbaty AAEE. LINEAR DIOPHANTINE GRAPHS. JAEM. 2026;16(6):771-785. https://izlik.org/JA78RZ84ZA

Chicago

Nasr, Amr, Mohamed Anwar, Mohammed Abd-El Azim Seoud, and Ahmed Abd Elgawad Elawady Elsonbaty. 2026. “LINEAR DIOPHANTINE GRAPHS”. TWMS Journal of Applied and Engineering Mathematics 16 (6): 771-85. https://izlik.org/JA78RZ84ZA.

EndNote

Nasr A, Anwar M, Seoud MA-EA, Elsonbaty AAEE (June 1, 2026) LINEAR DIOPHANTINE GRAPHS. TWMS Journal of Applied and Engineering Mathematics 16 6 771–785.

IEEE

[1]A. Nasr, M. Anwar, M. A.-E. A. Seoud, and A. A. E. E. Elsonbaty, “LINEAR DIOPHANTINE GRAPHS”, JAEM, vol. 16, no. 6, pp. 771–785, June 2026, [Online]. Available: https://izlik.org/JA78RZ84ZA

ISNAD

Nasr, Amr - Anwar, Mohamed - Seoud, Mohammed Abd-El Azim - Elsonbaty, Ahmed Abd Elgawad Elawady. “LINEAR DIOPHANTINE GRAPHS”. TWMS Journal of Applied and Engineering Mathematics 16/6 (June 1, 2026): 771-785. https://izlik.org/JA78RZ84ZA.

JAMA

1.Nasr A, Anwar M, Seoud MA-EA, Elsonbaty AAEE. LINEAR DIOPHANTINE GRAPHS. JAEM. 2026;16:771–785.

MLA

Nasr, Amr, et al. “LINEAR DIOPHANTINE GRAPHS”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 6, June 2026, pp. 771-85, https://izlik.org/JA78RZ84ZA.

Vancouver

1.Amr Nasr, Mohamed Anwar, Mohammed Abd-El Azim Seoud, Ahmed Abd Elgawad Elawady Elsonbaty. LINEAR DIOPHANTINE GRAPHS. JAEM [Internet]. 2026 Jun. 1;16(6):771-85. Available from: https://izlik.org/JA78RZ84ZA