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On the gcd-graphs over polynomial rings

Part of: Graph theory

Published online by Cambridge University Press:  01 October 2025

Ján Mináč ,
Tung T. Nguyen  and
Nguyễn Duy Tân
Ján Mináč
Affiliation:
Western University (University of Western Ontario) , Canada e-mail: minac@uwo.ca
Tung T. Nguyen
Affiliation:
Lake Forest College , United States e-mail: tnguyen@lakeforest.edu
Nguyễn Duy Tân*
Affiliation:
Faculty Mathematics and Informatics, Hanoi University of Science and Technology , Vietnam
*
e-mail: tan.nguyenduy@hust.edu.vn
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Abstract

Gcd-graphs over the ring of integers modulo n are a natural generalization of unitary Cayley graphs. The study of these graphs has foundations in various mathematical fields, including number theory, ring theory, and representation theory. Using the theory of Ramanujan sums, it is known that these gcd-graphs have integral spectra; i.e., all their eigenvalues are integers. In this work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We establish some fundamental properties of these graphs, emphasizing their analogy to their counterparts over ${\mathbb {Z}}.$

Keywords

Gcd-graphsCayley graphsperfect graphsconnected graphsbipartite graphs

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 05C51: Graph designs and isomomorphic decomposition

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Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 28
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

J.M. is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant R0370A01. He gratefully acknowledges the Western University Faculty of Science Distinguished Professorship 2020–2021. T.T.N. is partially supported by an AMS-Simons Travel Grant. Parts of this work were done while he was a Postdoc Associate at Western University. He thanks Western University for their hospitality. N.D.T. is partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2023.21.

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