https://doi.org/10.30598/barekengvol17iss2pp0667-0672 June 2023 Volume 17 Issue 2 Page 0667–0672 P-ISSN: 1978-7227 E-ISSN: 2615-3017 BAREKENG: Journal of Mathematics and Its Applications 667 BIPARTITE GRAPH ASSOCIATED WITH ELEMENTS AND COSETS OF SUBRINGS OF FINITE RINGS Hubbi Muhammad 1* , Niswah Qonita 2 , R. A. Wahyu Fibriyanti 3 , Yeni Susanti 4 1,2,3,4 Department of Mathematics, Faculty of Mathematics and Natural Science, Gadjah Mada University Bulaksumur, Depok, Sleman, Special Region of Yogyakarta, 55281, Indonesia Corresponding author’s e-mail: * hubbimuhammad@mail.ugm.ac.id ABSTRACT Article History: Let  be a finite ring. The bipartite graph associated to elements and cosets of subrings of  is a simple undirected graph Γ() with vertex set  ∪   , where   is the set of all subrings of , and two vertices  ∈  and ∈  are adjacent if and only if  = . In this study, we investigate some basic properties of the graph Γ(). In particular, we investigate some properties of Γ( 2 (ℤ  )), where  2 (ℤ  ) is the ring of matrices over ℤ  . Also, we study the diameter of the bipartite graph associated to the quaternion ring. Received: 29 th September 2022 Revised: 7 th April 2023 Accepted: 10 th April 2023 Keywords: Bipartite graph; Cosets; Finite ring; Matrices; Quaternion This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-ShareAlike 4.0 International License. How to cite this article: H. Muhammad, N. Qonita, R. A. W. Fibriyanti and Y. Susanti., “BIPARTITE GRAPH ASSOCIATED WITH ELEMENTS AND COSETS OF SUBRINGS OF FINITE RINGS,” BAREKENG: J. Math. & App., vol. 17, iss. 2, pp. 0667-0672, June, 2023. Copyright © 2023 Author(s) Journal homepage: https://ojs3.unpatti.ac.id/index.php/barekeng/ Journal e-mail: barekeng.math@yahoo.com; barekeng.journal@mail.unpatti.ac.id Research Article • Open Access