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All Journal Electronic Journal of Graph Theory and Applications (EJGTA) Indonesian Journal of Mathematics and Natural Sciences Sainteknol : Jurnal Sains dan Teknologi International Journal of Advances in Intelligent Informatics Educational Management Humanis : Jurnal Ilmu-Ilmu Sosial dan Humaniora JRPM (Jurnal Review Pembelajaran Matematika) PROFIT JTAM (Jurnal Teori dan Aplikasi Matematika) Unnes Journal of Mathematics Education Research Journal of Primary Education Symmetry: Pasundan Journal of Research in Mathematics Learning and Education M A T H L I N E : Jurnal Matematika dan Pendidikan Matematika Imajiner: Jurnal Matematika dan Pendidikan Matematika SEPREN: Journal of Mathematics Education and Applied Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi Journal Evaluation in Education (JEE) Unnes Journal of Mathematics Proceeding ISETH (International Summit on Science, Technology, and Humanity) SJME (Supremum Journal of Mathematics Education)
Isnaini Rosyida, Isnaini
Department Of Mathematics, Universitas Negeri Semarang, Semarang, Indonesia
Humanities Computer Science & IT Decision Sciences, Operations Research & Management Economics, Econometrics & Finance Education Electrical & Electronics Engineering Engineering Languange, Linguistic, Communication & Media Mathematics Medicine & Pharmacology Social Sciences Other

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Journal : Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi

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Penentuan Spektrum pada Variasi Graf Barbel Putri, Neli Septiana; Rosyida, Isnaini
Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi Volume 13 Issue 3 December 2025
Publisher : Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.37905/euler.v13i3.33968

Abstract

This study aims to analyze the determination of the spectrum of barbell graph variations, where the variations are made by modifying the number of nodes on the bridge between complete graphs in a barbell structure. The spectrum contains the eigenvalues of the adjacency matrix of the barbell graph variations along with their multiplicities. The analysis is conducted manually using linear algebra approaches such as cofactor expansion, characteristic polynomial factorization, the rational root theorem, and Horner’s scheme. The results are then validated using Python programming. The findings of this study show that the longer and more complex the bridge connecting the two complete graphs, the greater the diversity of eigenvalues produced. The spectrum of the barbell graph B(n,1)B(n, 1)B(n,1) consists of the eigenvalues λ1,n−1,λ2,−1,λ3\lambda_1, n - 1, \lambda_2, -1, \lambda_3λ1,n−1,λ2,−1,λ3 with multiplicities 1,1,1,2n−3,11, 1, 1, 2n - 3, 11,1,1,2n−3,1. Furthermore, the spectrum of the barbell graph B(n,2)B(n, 2)B(n,2) consists of the eigenvalues λ1,λ2,λ3,λ4,−1,λ5,λ6\lambda_1, \lambda_2, \lambda_3, \lambda_4, -1, \lambda_5, \lambda_6λ1,λ2,λ3,λ4,−1,λ5,λ6 with multiplicities 1,1,1,1,2n−4,11, 1, 1, 1, 2n - 4, 11,1,1,1,2n−4,1, respectively. This research provides theoretical contributions regarding the relationship between complex graph structures and their spectral representations.
Co-Authors - Hwihanus Abdurahim, Abdurahim Adi Nur Cahyono Aini, Aulia Hurul Anggraini, Lana Aristya Arifah, Nafiqotun Asih, Tri Sri Noor Aula, Maulida Fatma Reza Baihaqi, Muhamad Adzib Budi Waluya, St. Danieryanto, Erlia Dewi, Nuraliftsa Maharani Diari Indriati Fitriyani, Eka Retna Hardi Suyitno Have Zulkarnaen Huda, Muhamad Nurul Indah E. Wijayanti Isnarto, Isnarto Iwan junaedi Jahfaludin, Abi Junaedi , Iwan Kharomah, Siti Ismiatul Kiki A. Sugeng Kristina Wijayanti Mashuri Mashuri Mulyono Mulyono Nisarohmah, Nur Ilmia Noriza Munahefi, Detalia Novi Darmayanti Nurdini, Eka Nurhalisha, Erina Nurkaromah Dwidayati, Nurkaromah Priscilia, Priscilia Monica Praneswari Putri, Neli Septiana Putriaji Hendikawati Rachmani Dewi, Nuriana Radian Sri Rama Riawati, Beti Riyadhotul Janah, Siti Rochmad - Safitri, Fristy Aulia Salsadina, Nida Alifa Scolastika Mariani Siti Muawanah Siti Nur Hidayati Slamet Riyadi St. Budi Waluya Suparmi Suparmi Suryonegoro, Bayu Murti Suryono Suryono Susanti, Desmiani Sutri Handayani W. Widodo Wahyuningsih, Vebryana Wardono Wardono Wibowo, Harris Adi Widiantomo, Gilar Ajie Wiharno, Wiharno Zaenuri Mastur