Energi Detour pada Graf Invers dari Grup Quaternion Diperumum

Iftitahur Rohmah, Mohammad Nafie Jauhari, Erna Herawati

Abstract


A generalized quaternion group  is a non-abelian group with a  order constructed from the two elements  that is denoted by  defined as   where the  is the identity, . The inverse graph of a group is a graph which the vertices set are all elements of a finite group and two distinct vertices  and  are adjacent if and only if either  or . Graph energy, especially detour energy, is an important aspect in graph theory that describes network stability and resistance. The method used in this research is a qualitative method. This study aims to determine the formula of the detour energy in the inverse graph of the quaternion group is announced with . Through algebra analysis methods and graph theory, it is found that detour energy can be calculated using the detour matrix and the resulting eigen value. The results showed that the detour energy formula in the inverse graph from the quaternion group was founded was  for each .


Keywords


Generalized Quaternion Group; Inverse Graph; Detour Energy

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References


Abdussakir, N. N. Azizah, and F. F. Nofandika, Teori Graf: Topik dasar untuk tugas akhir/skripsi. 2009.

U. Jannah et al., “Indeks jumlah jarak eksentrik graf invers dari grup quaternion diperumum,” 2023.

J. A. Gallian, “contemporary abstract algebra,” Ninth edit.,

D. S. Dummit and R. M. Foote, Abstract_algebra_Third_Edition_Foote_Dum.pdf, Third edit. John wiley & Sons, Inc, 2004.

X. L. Ma, H. Q. Wei, and G. Zhong, “The Cyclic Graph of a Finite Group,” Algebra, vol. 2013, pp. 1–7, 2013, doi: 10.1155/2013/107265.

R. B. Bapat, Graphs and matrices. 2010. doi: 10.1017/cbo9780511659843.006.

G. Chartrand, L. Lesniak, and P. Zhang, Graphs & digraphs. 2010. doi: 10.1201/b19731.

M. R. Alfuraidan and Y. F. Zakariya, “Inverse graphs associated with finite groups,” Electron. J. Graph Theory Appl., vol. 5, no. 1, pp. 142–154, 2017, doi: 10.5614/ejgta.2017.5.1.14.

A. Abdussakir, “Detour Energy of Complement of Subgroup Graph of Dihedral Group,” ZERO J. Sains, Mat. dan Terap., vol. 1, no. 2, pp. 41–48, 2017, doi: 10.30829/zero.v1i2.1460.

I. Gutman, S. Z. Firoozabadi, J. A. De La Peña, and J. Rada, “On the energy of regular graphs,” Match, vol. 57, no. 2, pp. 435–442, 2007.

K. Erciyes, “Graph-Theoretical Analysis of Biological Networks: A Survey,” 2023, doi: 10.20944/preprints202307.1012.v1.

Larson, Edwards, and Falvo, Elementary linear algebra. Cengage Learning, Sixth edit. 2009.

R. Balakrishnan, “The energy of a graph,” Linear Algebra Appl., vol. 387, no. 1-3 SUPPL., pp. 287–295, 2004, doi: 10.1016/j.laa.2004.02.038.

K. C. Das and S. A. Mojallal, “On Laplacian energy of graphs,” Discrete Math., vol. 325, no. 1, pp. 52–64, 2014, doi: 10.1016/j.disc.2014.02.017.

S. K. Ayyaswamy and S. Balachandran, “On detour spectra of some graphs,” World Acad. Sci. Eng. Technol., vol. 67, no. 7, pp. 529–531, 2010.




DOI: https://doi.org/10.18860/jrmm.v4i2.30060

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