AN ANALYSIS OF THE SEIDEL LAPLACIAN ENERGY OF A FUZZY INTUITIONISTIC SYSTEM
DOI:
https://doi.org/10.55766/sujst7197Keywords:
Energy of a graph, Fuzzy graph, Intuitionistic fuzzy graph, Laplacian energy, Seidel Laplacian energyAbstract
This paper presents one of the latest research works in intuitionistic fuzzy graph theory. Along with questionable concepts conveyed in distinctive languages, intuitionistic fuzzy set theory offers a noteworthy and ground-breaking depiction of vulnerability estimation. The concept of energy is related to the spectrum of a graph. The energy of graphs plays a vital role in graph theory. In mathematics, the total sum of the absolute values of the eigenvalues of the graph’s adjacency matrix is referred to as the graph’s energy. In the framework of spectral graph theory, this quantity is extensively studied. When analyzing various repulsive and appealing circumstances, energy graphs are a great tool. Analogous energies based on the eigenvalues of various additional graph matrices have been examined recently. Graph energy possesses numerous mathematical characteristics which are investigated in this paper. One type of graph energy is Seidel Laplacian energy. The Seidel Laplacian energy (SLE) of a graph has already been extended to fuzzy graphs. In this manuscript, it is extended to an intuitionistic fuzzy graph. Energy, Laplacian energy, and SLE of an intuitionistic fuzzy graph are evaluated using some examples, and their lower and upper bounds are derived.
References
Akram, M., and Akmal, R. (2016). Operations on intuitionistic fuzzy graph structures. Fuzzy Information and Engineering, 8(4):389-400. https://doi.org/10.1016/j.fiae.2017.01.001
Askari, J. (2023). A note on the Seidel and Seidel Laplacian matrices. Boletim da Sociedade Paranaense de Matemática, 41:1-6. https://doi.org/10.5269/bspm.51593
Askari, J., Das, K.C., and Shang, Y. (2024). On the eigenvalues and energy of the Seidel and Seidel Laplacian matrices of graphs. Discrete Dynamics in Nature and Society, 2024, 8390307. https://doi.org/10.1155/2024/8390307
Atanosov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1):87-96. https://doi.org/10.1016/S0165-0114(86)80034-3
Balakrishnan, R. (2004). The energy of a graph. Linear Algebra and Its Applications, 387:287-295. https://doi.org/10.1016/j.laa.2004.02.038
Basha, S.S., and Kartheek, E. (2015). Laplacian energy of an intuitionistic fuzzy graph. Indian Journal of Science and Technology, 8(33):1-7. https://doi.org/10.17485/ijst/2015/v8i33/79899
Bhat, P.G., and D'Souza, S. (2017). Color signless Laplacian energy of graphs. AKCE International Journal of Graphs and Combinatorics, 14(2):142-148. https://doi.org/10.1016/j.akcej.2017.02.003
Bondy, J.A., and Murthy, S.R. (1976). Graph theory with applications. North Holland. https://doi.org/10.1007/978-1-349-03521-2
Deepa, G., and Chandrasekaran, V.M. (2018). Lower and upper bound of the Laplacian energy with real and complex roots of an intuitionistic fuzzy directed graph. International Journal of Applied Systemic Studies, 8(3). https://doi.org/10.1504/IJASS.2018.096097
Divya, N.G., Saju, M., Roshna, A., and Thomas, K.R. (2024). Smart nanomaterials in energy applications. Tailored Functional Materials for Clean and Sustainable Development, 1-18. https://doi.org/10.1201/9781003394761-22
D'Souza, S., Gowtham, H.J., Bhat, P.G., and Girija, K.P. (2022). Color Laplacian energy of some cluster graphs. TWMS Journal of Applied and Engineering Mathematics, 12(1):247-259.
Gutman, I. (1978). The energy of a graph. Ber. Math. Stat. Sekt. Forschungsz. Graz, 103:1-22.
Gutman, I., and Zhou, B. (2006). Laplacian energy of graphs. Linear Algebra and Its Applications, 414(1):29-37. https://doi.org/10.1016/j.laa.2005.09.008
Harary, F. (1969). Graph theory. Addison-Wesley. https://doi.org/10.21236/AD0705364
Harishchandra, S., Ramane, K., Ashoka, B., Patil, P.D. (2023). Some relations between energy and Seidel energy of a graph. Acta Univ. Sapientiae Informatica, 15(1):46-59. https://doi.org/10.2478/ausi-2023-0005
Hemalatha, R., and Somasundaram, K. (2022). Sombor index of edge corona product of some classes of graphs. South East Asian Journal of Mathematics and Mathematical Science, 18(3):307-316. https://doi.org/10.56827/SEAJMMS.2022.1803.25
Hussein, H.S. (2023). The state of the art of nanomaterials and its applications in energy saving. Bulletin of the National Research Centre, 47(1):7. https://doi.org/10.1186/s42269-023-00984-4
Jansirani, M.M., Lakshmanan, K., Kumar, L.S., Sivaranjani, K., Kulandaivelu, R., Santhiya, S., Antony Raj, A.S., and Vinodhini, D. (2024). Exploration of NαIg-closed and NαIg-open sets in nano ideal topological spaces with practical applications. Communications on Applied Nonlinear Analysis, 31(5):390-397. https://doi.org/10.52783/cana.v31.1058
Karunambigai, M.G., and Parvathi, R. (2006). Intuitionistic fuzzy graphs. Journal of Computational Intelligence, Theory and Applications, 20:139-150.
Kavitha, S., and Jayalalitha, G. (2023). Gaussian graceful labeling. European Chemical Bulletin, 12(Special issue 4), 8438-8456.
Kavitha, S., Jayalalitha, G., and Sivaranjani, K. (2024). Enhanced security in public key cryptography: A novel approach combining Gaussian graceful labeling and NTRU public key cryptosystem. EAI Endorsed Transactions on Internet of Things, 10:1-7. https://doi.org/10.4108/eetiot.4992
Kuppusamy, V., Shanmugasundaram, M., Bharathi, P.D., Martin-Barreiro, C., Cabezas, X., Leiva, V., and Castro, C. (2024). Addressing a decision problem through a bipolar Pythagorean fuzzy approach: A novel methodology applied to digital marketing. Heliyon, 10(3):e23991. https://doi.org/10.1016/j.heliyon.2024.e23991
Li, X., Shi, Y., and Gutman, I. (2012). Graph energy. Springer. https://doi.org/10.1007/978-1-4614-4220-2
Meenakshi, S., and Lavanya, S. (2014). A survey on energy of graphs. Annals of Pure and Applied Mathematics, 8(2):183-191.
Naggor, G.A., and Shajitha, B.S. (2010). Degree, order, and size in intuitionistic fuzzy graph. International Journal of Algorithm Computing and Mathematics, 3(1):11-16.
Narayanan, A., and Mathew, S. (2013). Energy of a fuzzy graph. Annals of Fuzzy Mathematics and Informatics, 6(3):455-465.
Oboudi, M.R. (2016). Energy and Seidel energy of graphs. Match Communication in Mathematical and Computer Chemistry, 75:291-303.
Praba, B., and Chandrasekaran, V.M. (2014). Energy of an intuitionistic fuzzy graph. Italian Journal of Pure and Applied Mathematics, 32:431-444.
Ramane, H.S., Jummannaver, R.B., and Gutman, I. (2017). Seidel Laplacian energy of graphs. International Journal of Applied Graph Theory, 1:74-82.
Ramesh, O., and Basha, S.S. (2020). The implementation of cosine similarity measures in decision-making problems by signless Laplacian energy of an intuitionistic fuzzy graph. European Journal of Molecular & Clinical Medicine, 7(10):239-251.
Reddy, N.R., Khan, M.Z., Basha, S.S., Alahmadi, A., Alahmadi, A.H., and Chowdhary, C.L. (2023). The Laplacian energy of hesitancy fuzzy graphs in decision-making problems. Computer Systems Science & Engineering, 44(3):2637-2653. https://doi.org/10.32604/csse.2023.029255
Shigehalli, V.S., and Betageri, S.K. (2015). Color Laplacian energy of graphs. Journal of Computer and Mathematical Sciences, 6(9):485-494.
Sivaranjani, K., Shanmuga Sundaram, O.V., and Akalyadevi, K. (2024). Seidel Laplacian energy of fuzzy graphs. EAI Endorsed Transactions on Energy Web, 11:1-6. https://doi.org/10.4108/ew.5297
Umapathy, K., Palanivelu, B., Leiva, V., Bharathi, P. D., and Castro, C. (2023). On fuzzy and crisp solutions of a novel fractional pandemic model. Fractal and Fractional, 7:528. https://doi.org/10.3390/fractalfract7070528
Yalcin, N.F. (2022). On Seidel Laplacian matrix and energy of graphs. Acta Univ. Sapientiae Informatica, 14(1):104-118. https://doi.org/10.2478/ausi-2022-0007
Zadeh, L.A. (1965). Fuzzy sets. Information Science, 8:338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
