Penentuan Spektrum pada Variasi Graf Barbel

Neli Septiana Putri; Isnaini Rosyida

doi:10.37905/euler.v13i3.33968

Penentuan Spektrum pada Variasi Graf Barbel

Neli Septiana Putri, Isnaini Rosyida

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.

Keywords

Spectrum; Barbel Graph Variation; Linear Algebra; Eigenvalues; Adjacency Matrix; Characteristic Polynomial

Full Text:

PDF

References

O. Jones, “Spectra of Simple Graphs,” pp. 1–20, May 2013.

M. Maulana and N. W. Switrayni, “Banyak Pohon Pembangun pada Graf Barbell,” EIGEN Math. J., vol. 2, no. 2, pp. 125–130, 2019, doi: 10.29303/emj.v1i2.46.

S. Eka R. and B. Rahadjeng, “Dimensi Metrik pada Graf Lintasan, Graf Komplit, Graf Sikel, Graf Bintang, dan Graf Bipartit Komplit,” Mathunesa J. Ilm. Mat., vol. 3, no. 1, pp. 1–6, 2014.

I. K. Budayasa, Teori Graf dan Aplikasinya. Surabaya: Unesa University Press, 2007.

Y. Agustina, N. Kusumastuti, and F. Fran, “Spektrum Adjacency dari Graf Bintang, Graf Mahkota, dan Graf Tangga,” Bul. Ilm. Math. Stat. dan Ter., vol. 11, no. 2, pp. 355–362, 2022, doi: 10.26418/bbimst.v11i02.53668.

R. Oktriyansa, Triyani, and S. R. Nurshiami, “Spektrum Laplace pada Graf Bipartit Lengkap (Kn,n),” Pros. Semin. Nas. Mat. dan Ter., vol. 3, pp. 51–64, 2022.

O. Kholifah, S. Rahmah Nurshiami, and S. Maryani, “Spektrum Detour dan Indeks Detour dari Graf Berlian,” Pros. Semin. Nas. Mat. dan Ter., vol. 3, pp. 103–116, 2022.

H. Komarullah, “The Adjacency Spectrum of Triangular Book Graphs,” Proc. Int. Conf. Educ. Sharia, vol. 1, no. 1, pp. 89–93, 2024, doi: 10.62097/ices.v124.15.

Karmilawati, “Spektrum Detour pada Graf Helm Tertutup,” Bul. Ilm. Math. Stat. dan Ter., vol. 12, no. 6, pp. 561–568, 2023, doi: 10.26418/bbimst.v12i6.72387.

M. Abdy, R. Syam, and A. M. Putri, “Spectrum Matriks Detour dari Graf Roda dengan n + 1 Titik Wn,” J. Math. Comput. Stat., vol. 3, no. 1, pp. 32–40, 2020. [Online]. Available: http://www.ojs.unm.ac.id/jmathcos

R. Mulyani, Triyani, and N. Larasati, “Spektrum pada Graf Reguler Kuat,” JMP, no. 1, pp. 13–21, 2013, doi: 10.20884/1.jmp.2013.5.1.2912.

I. Sason, N. Krupnik, S. Hamud, and A. Berman, “On Spectral Graph Determination,” Mathematics, vol. 13, no. 4, p. 549, 2025, doi: 10.3390/math13040549.

Y. C. Sewar, G. L. Putra, F. O. Haning, and I. G. Pasangka, “Spektrum Matriks Ketetanggaan Graf Cayley pada Grup Zn,” J. Ris. Ap. Mat., vol. 9, no. 1, pp. 26–43, 2025, doi: 10.26740/jram.v9n1.

K. Kumari and P. Panigrahi, “Universal Adjacency Spectrum of (Proper) Power Graphs and Their Complements on Some Groups,” arXiv, pp. 1–27, 2023. Available: https://arxiv.org/abs/2307.09129.

T. Triyani, R. Oktriyansa, and S. R. Nurshiami, “Spektrum Laplace pada Graf Mahkota dan Graf Bentang,” Pythagoras: J. Mat. P. Mat., vol. 19, no. 1, pp. 27–36, 2024, doi: 10.21831/pythagoras.v19i1.60818.

T. Alfin, W. B. Kawuwung, and T. Tandianganga, “Spektrum Matriks Detour Graf Teratur,” SAINS: J. MIPA dan Pengajarannya, vol. 23, no. 2, pp. 41–45, 2023, doi: 10.31957/sains.v23i1.3254.

DOI: https://doi.org/10.37905/euler.v13i3.33968