Sains Malaysiana 53(6)(2024): 1421-1426

http://doi.org/10.17576/jsm-2024-5306-15

 

Degree Square Subtraction Energy of Non-Commuting Graph for Dihedral Groups 

(Tenaga Tolak Darjah Kuasa Dua bagi Graf Tak Kalis Tukar Tertib untuk Kumpulan Dwihedron)

 

MAMIKA UJIANITA ROMDHINI1,*, ATHIRAH NAWAWI2, FAISAL AL-SHARQI3,4,* & MUHAMMAD RIJAL ALFIAN1                 

1Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Mataram, Mataram 83125, Indonesia 

2Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

3Department of Mathematics, Faculty of Education for Pure Sciences, University Of Anbar, Ramadi, Anbar, Iraq

4College of Engineering, National University of Science and Technology, Dhi Qar, Iraq

 

Received: 10 October 2023/Accepted: 26 April 2024


Abstract

The non-commuting graph on a finite , denoted by , with the set of non-central elements of as the vertex set and two distinct vertices are adjacent whenever they do not commute in . In this paper, we discuss the spectrum, spectral radius and degree square subtraction energy of  for dihedral groups of order ,  where . It is found that the obtained energy here is equal to twice its spectral radius and there is a relationship with the degree subtraction energy that was described in previous literature.

 

Keywords: Degree square subtraction matrix; dihedral group; non-commuting graph; the energy of a graph         

Abstrak

Graf tak kalis tukar tertib ditakrifkan pada suatu kumpulan terhingga , ditandakan dengan , dengan set unsur bukan pusat  sebagai set bucu dan dua bucu berbeza adalah bersebelahan apabila mereka tak kalis tukar tertib dalam . Dalam makalah ini, kita membincangkan spektrum, jejari spektrum dan tenaga tolak darjah kuasa dua bagi  untuk kumpulan dwihedron peringkat , , yang . Didapati bahawa tenaga yang diperoleh ini adalah sama dengan dua kali jejari spektrumnya dan terdapat hubungan dengan tenaga tolak darjah yang telah diterangkan dalam kajian terdahulu.

 

Kata kunci: Graf tak kalis tukar tertib; kumpulan dwihedron; matriks tolak darjah kuasa dua; tenaga graf

 

REFERENCES

Abdollahi, A., Akbari, S. & Maimani, H.R. 2006. Non-commuting graph of a group. J. Algebra. 298(2): 468-492.

Akram, M. & Naz, S. 2018. Energy of Pythagorean fuzzy graphs with applications. Mathematics 6(8): 136.

Angadi, S.A. & Hatture, S.M. 2019. Face recognition through symbolic modelling of face graphs and texture. International Journal of Pattern Recognition and Artificial Intelligence 33(12): 1956008.

Ankayarkanni & Leni, E.S. 2014. A technique for classification of high resolution satellite images using object-based segmentation. Journal of Theoretical and Applied Information Technology 68: 275-286.

Aschbacher, M. 2000. Finite Group Theory. Cambridge: University Press.

Bapat, R.B. & Pati, S. 2004. Energy of a graph is never an odd integer. Bull. Kerala. Math. Assoc. 1: 129-132.

Cameron, P.J. 2023. What can graphs and algebraic structures say to each other? ACKE International Journal of Graphs and Combinatoricsdoi:10.1080/09728600.2023.2290036.

Daianu, M., Mezher, A., Jahanshad, N., Hibar, D.P., Nir, T.M., Jack, C.R., Weiner, M.W., Bernstein, M.A. & Thompson, P.M. 2015. Spectral graph theory and graph energy metrics show evidence for the alzheimer’s disease disconnection syndrome in APOE-4 risk gene carriers. IEEE International Symposium on Biomedical Imaging. 2015: 458-461.

Dasgupta, A., Das, R., Nayak, L. & De, R.K. 2015. Analysing epileptogenic brain connectivity networks using clinical EEG data. IEEE International Conference on Bioinformatics and Biomedicine 2015: 815-821.

Dhanalakshmi, A., Rao, K.S. & Sivakumar, K. 2015. Characterization of α-cyclodextrin using adjacency and distance matrix. Indian Journal of Science 12: 78-83.

Gutman, I. 1978. The energy of graph.  Ber. Math. Statist. Sekt. Forschungszenturm Graz. 103: 1-22.

Gutman, I. & Furtula, B. 2019. Graph energies and their applications. Bulletin T. CLII de l’Acad´emie serbe des sciences et des arts, Classe des Sciences math´ematiques et naturelles Sciences math´ematiques 44: 2-45.

Horn, R.A. & Johnson, C.A. 1985. Matrix Analysis. Cambridge: Cambridge University Press.

Huang, C.H., Tsai, J.J.P., Kurubanjerdjit, N. & Ng, K.L. 2019. Computational analysis of molecular networks using spectral graph theory, complexity measures and information theory. BioArxiv 1-39. https://doi.org/10.1101/536318

Jiang, J., Zhang, R., Guo, L., Li, W. & Cai, X. 2016. Network aggregation process in multilayer air transportation networks. Chinese Physics. Letter 33: 108901.

Khasraw, S.M.S., Ali, I.D. & Haji, R.R. 2020. On the non-commuting graph for dihedral group. Electron. J. Graph Theory Appl. 8(2): 233-239.

Li, X., Shi, Y. & Gutman, I. 2012. Graph Energy. New York: Springer.

Macha, J.S. & Shinde, S. 2022. Degree square subtraction spectra and energy. J. Indones. Math. Soc. 28(3): 259-271.

Musulin, E. 2014. Spectral graph analysis for process monitoring. Industrial & Engineering Chemistry Research 53: 10404-10416.

Pirzada, S. & Gutman, I. 2008. Energy of a graph is never the square root of an odd integer. Appl. Anal. Discr. Math. 2: 118-121.

Praba, B., Deepa, G. & Chandrsekaran, V.M. 2016. Spreading rate of virus between incoming and outgoing links of a website through an intuitionistic fuzzy graph. International Journal of Pure and Applied Mathematics 109: 799-812.

Pugliese, A. & Nilchiani, R. 2017. Complexity analysis of fractionated spacecraft architectures. American Institute of Aeronautics and Astronautics Space Forum 2017: 2721275.

Ramane, H.S. & Shinde, S.S. 2017. Degree exponent polynomial of graphs obtained by some graph operations. Electron. Notes Discrete Math. 63: 161-168.

Romdhini, M.U., Nawawi, A. & Chen, C.Y. 2023. Neighbors degree sum energy of commuting and non-commuting graphs for dihedral groups. Malaysian J. Math. Sci. 17(1): 53-65.

Romdhini, M.U., Nawawi, A. & Chen, C.Y. 2022. Degree exponent sum energy of commuting graph for dihedral groups. Malaysian J. Sci. 41(sp1): 40-46.

Romdhini, M.U. & Nawawi, A. 2023. Degree subtraction energy of commuting and non-commuting graphs for dihedral groups. Int. J. Math. Comput. Sci. 18(3): 497-508.

Romdhini, M.U. & Nawawi, A. 2022a. Degree sum energy of non-commuting graph for dihedral groups. Malaysian J. Sci. 41(sp1): 34-39.

Romdhini, M.U. & Nawawi, A. 2022b. Maximum and minimum degree energy of commuting graph for dihedral groups. Sains Malaysiana 51(12): 4145-4151.

Singh, T., Baths, V. & Kumar, A. 2014. Disruption of cell wall fatty acid biosynthesis in Mycobacterium tuberculosis using the concept of minimum robust domination energy of graph. Annual Research & Review in Biology 4: 2037-2044.

Sinha, K. & Suh, E.S. 2018. Pareto-optimization of complex system architecture for structural complexity and modularity. Research in Engineering Design 29: 123-141.

Sun, D., Xu, C. & Zhang, Y. 2016. A novel method of 2D graphical representation for proteins and its application. MATCH Communications in Mathematical and in Computer Chemistry 75: 431-446.

Van Mieghem, P. & van de Bovenkamp, R. 2015. Accuracy criterion for the mean–field approximation in susceptible-infected-susceptible epidemics on networks. Physical Review E 91: 032812.

Xiao, B., Song, Y.Z. & Hall, P. 2011. Learning invariant for object identification by using graph methods. Computer Vision and Image Understanding 115: 1023-1031.

Yuge, K. 2018. Extended configurational polyhedra based on graph representation for crystalline solids. Transactions of the Materials Research Society of Japan 43: 233-236.

Zhang, H., Bai, X., Zheng, H., Zhao, H., Zhou, J., Cheng, J. & Lu, H. 2013. Hierarchical remote sensing image analysis via graph Laplacian energy. IEEE  Geoscience and Remote Sensing Letters 10: 396-400.

 

*Corresponding author; email: mamika@unram.ac.id

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
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