SACS | Modified Sombor Spectral Radii and Modified Sombor Energies of Splitting and Shadow Graphs

Published in Volume XXXIV, Issue 2, 2024, pages 89-112, doi: 10.47743/SACS.2024.2.89

Authors: A. Bilal, M. Mobeen Munir

Abstract

Gutman et al. introduced Sombor and Modified Somber index because of the rapidly growing applications in chemistry and network analysis. Graph energy ε(G) and the spectral radius ℘(G) of graph G are essential components that are associated with the eigenvalues of the matrix of graph G and, chemically, with the intermolecular forces. These graph invariants have many useful applications in computer sciences, networking, and molecular computing. There are numerous variants of the ε(G) and ℘(G) attained by substituting another matrix in place of an adjacency matrix. Modified Sombor energy, MSε(G) is defined as the sum of absolute eigenvalues of the modified Sombor matrix or in other words we can say that modified Sombor energy, MSε(G) is the trace norm of modified Sombor matrix. Modified Sombor spectral radius, ℘MS is defined as the largest absolute eigenvalue of the modified Sombor matrix. The major focus of this article is on the MSε(G), ℘MS of the generalized shadow and splitting graphs. The only realistic problem in which we are particularly interested in how MSε(Spl_t(G)) and MSε(Sh_t(G)) are comparable to MSε(G). On similar lines we are also interested in how ℘MS(Spl_t(G)) and ℘MS(Sh_t(G)) are comparable to ℘MS(G). We were able to address these challenges by focusing on splitting and shadow graphs.

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References

[1] Dhanalakshmi Amaresan and Srinivasa Rao Konda. Characterization of α-cyclodextrin using adjacency and distance matrix. Indian Journal of Science, 12(35):78–83, 2015.

[2] Alexandru T. Balaban. Applications of graph theory in chemistry. Journal of Chemical Information and Computer Sciences, 25(3):334–343, 1985. doi: 10.1021/CI00047A033.

[3] Ahmad Bilal and Muhammad Mobeen Munir. ABC energies and spectral radii of some graph operations. Frontiers in Physics, 10, 2022. doi:10.3 389/fphy.2022.1053038.

[4] Ahmad Bilal and Muhammad Mobeen Munir. Randic and reciprocal randic spectral radii and energies of some graph operations. Journal of Intelligent and Fuzzy Systems, 44(4):5719–5729, 2023. doi:10.3233/JIFS-221938.

[5] Ahmad Bilal and Muhammad Mobeen Munir. SDD spectral radii and SDD energies of graph operations. Theoretical Computer Science, 1007:114651, 2024. doi:10.1016/J.TCS.2024.114651.

[6] Ahmad Bilal, Muhammad Mobeen Munir, Muhammad Imran Qureshi, and Muhammad Athar. ISI spectral radii and ISI energies of graph operations. Frontiers in Physics, 11, 2023. doi:10.3389/fphy.2023.1149006.

[7] Dragoš M. Cvetković, Michael Doob, and Horst Sachs. Spectra of Graphs: Theory and Application. Pure and applied mathematics : a series of monographs and textbooks. Academic Press, 1980.

[8] Dragoš M. Cvetković and Peter Rowlinson. The largest eigenvalue of a graph: A survey. Linear and Multilinear Algebra, 28(1-2):3–33, 1990. doi: 10.1080/03081089008818026.

[9] Luisa Di Paola, Giampiero Mei, Almerinda Di Venere, and Alessandro Giuliani. Exploring the stability of dimers through protein structure topology. Current Protein & Peptide Science, 17(1):30–36, 2016. doi:10.2174/1389203716666150923104054.

[10] Feliks Ruvimovich Gantmakher. The Theory of Matrices, volume 1 of The Theory of Matrices. 1960.

[11] Alessandro Giuliani, Simonetta Filippi, and Marta Bertolaso. Why network approach can promote a new way of thinking in biology. Frontiers in Genetics, 5, 2014. doi:10.3389/fgene.2014.00083.

[12] Ivan Gutman. The energy of a graph. Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz, 103:1–22, 1978.

[13] Ivan Gutman. Geometric approach to degree–based topological indices: Sombor indices. MATCH Communications in Mathematical and in Computer Chemistr, 86:11–16, 2021.

[14] Özge Çlakoğlu Havare. On the inverse sum indeg index of some grap operations. Journal of the Egyptian Mathematical Society, 28, 2020. doi:10.1186/s42787-020-00089-1.

[15] Sakander Hayat and Muhammad Imran. Computation of topological indices of certain networks. Applied Mathematics and Computation, 240:213–228, 2014. doi:10.1016/J.AMC.2014.04.091.

[16] Sakander Hayat, Asad Khan, Muhammad Yasir Hayat Malik, Muhammad Imran, and Muhammad Kamran Siddiqui. Fault-tolerant metric dimension of interconnection networks. IEEE Access, 8:145435–145445, 2020. doi:10.1109/ACCESS.2020.3014883.

[17] Roger A. Horn and Charles R. Johnson. Topics in Matrix Analysis. Cambridge University Press, 1991. doi:10.1017/CBO9780511840371.

[18] Yufei Huang and Hechao Liu. Bounds of modified Sombor index, spectral radius and energy. AIMS Mathematics, 6(10):11263–11274, 2021. doi:10.3934/math.2021653.

[19] Muhammad Imran, Sakander Hayat, and Muhammad Yasir Hayat Mailk. On topological indices of certain interconnection networks. Applied Mathematics and Computation, 244:936–951, 2014. doi:10.1016/J.AMC.2014.07.064.

[20] Jian Jiang, Rui Zhang, Long Guo, Wei Li, and Xu Cai. Network aggregation process in multilayer air transportation networks. Chinese Physics Letters, 33:108901, 2016. doi:10.1088/0256-307X/33/10/108901.

[21] Asad Khan, Sakander Hayat, Yubin Zhong, Amina Arif, Laiq Zada, and Meie Fang. Computational and topological properties of neural networks by means of graph-theoretic parameters. Alexandria Engineering Journal, 66:957–977, 2023. doi:10.1016/j.aej.2022.11.001.

[22] V. Kulli and Ivan Gutman. Computation of Sombor indices of certain networks. International Journal of Applied Chemistry, 8:1–5, 01 2021. doi:10.14445/23939133/IJAC-V8I1P101.

[23] Xueliang Li, Yongtang Shi, and Ivan Gutman. Graph Energy. Springer New York, 2012. doi:10.1007/978-1-4614-4220-2.

[24] Igor Milovanović, Emina Milovanović, and Marjan Matejić. On some mathematical properties of Sombor indices. Bulletin of International Mathematical Virtual Institute, 11(2):341, 2021. doi:10.2298/JSC201215006R.

[25] Jure Pražnikar, Miloš Tomić, and Dušan Turk. Validation and quality assessment of macromolecular structures using complex network analysis. Scientific Reports volume, 9:1678, 2019. doi:10.1038/s41598-019-38658-9.

[26] Antonio Pugliese and Roshanak Nilchiani. Complexity analysis of fractionated spacecraft architectures. In AIAA SPACE and Astronautics Forum and Exposition, pages 1–9, 2017. doi:10.2514/6.2017-5118.

[27] Milan Randic. Characterization of molecular branching. Journal of the American Chemical Society, 97(23):6609–6615, 1975. doi:10.1021/ja00856a001.

[28] Hassan Raza, Sakander Hayat, and Xiang-Feng Pan. On the fault-tolerant metric dimension of certain interconnection networks. Journal of Applied Mathematics and Computing, 60(1-2):517–535, 2019. doi:10.1007/S12190-018-01225-Y.

[29] Izudin Redžepović. Chemical applicability of Sombor indices : Survey. Journal of the Serbian Chemical Society, 86(5):445–457, 2021. doi:10.2298/JSC201215006R.

[30] E. Sampathkumar and H.B. Walikar. On splitting graph of a graph. The Karnataka University Journal, 25-26(13):13–16, 1980-1981.

[31] Yilun Shang. Sombor index and degree-related properties of simplicial networks. Applied Mathematics and Computation, 419:126881, 2022. doi:10.1016/J.AMC.2021.126881.

[32] Hafiz Muhammad Afzal Siddiqui, Sakander Hayat, Asad Khan, Muhammad Imran, Ayesha Razzaq, and Jia-Bao Liu. Resolvability and fault-tolerant resolvability structures of convex polytopes. Theoretical Computer Science, 796:114–128, 2019. doi:10.1016/J.TCS.2019.08.032.

[33] J. J. Sylvester. Chemistry and algebra. Nature, 17:285, 1878. doi:10.1038/017284a0.

[34] Haiyan Wu, Yusen Zhang, Wei Chen, and Zengchao Mu. Comparative analysis of protein primary sequences with graph energy. Physica A: Statistical Mechanics and its Applications, 437:249–262, 2015. doi:10.1016/j.physa.2015.04.017.

[35] Lulu Yu, Yusen Zhang, Ivan Gutman, Yongtang Shi, and Matthias Dehmer. Protein sequence comparison based on physicochemical properties and the position-feature energy matrix. Scientific Reports, 7:46237, 2017. doi:10.1038/srep46237.

[36] Hong Yuan. Upper bounds of the spectral radius of graphs in terms of genus'. Journal of Combinatorial Theory, Series B, 74(2):153–159, 1998. doi:10.1006/JCTB.1998.1837.

[37] Koretaka Yuge. Extended configurational polyhedra based on graph representation for crystalline solids. Transactions of the Materials Research Society of Japan, 43(4):233–236, 2018. doi:10.14723/tmrsj.43.233.

[38] Xiujun Zhang, Ahmad Bilal, M. Mobeen Munir, and Hafiz Mutte ur Rehman. Maximum degree and minimum degree spectral radii of some graph operations. Mathematical Biosciences and Engineering, 19(10):10108–10121, 2022. doi:10.3934/mbe.2022473.

[39] Yusen Zhang, Chunrui Xu, and Yusen Zhan. Novel method of 2D graphical representation for proteins and its application. MATCH Communications in Mathematical and in Computer Chemistry, 75:431–446, 2016.

[40] Xuewu Zuo, Bilal Ahmad Rather, Muhammad Imran, and Akbar Ali. On some topological indices defined via the modified Sombor matrix. Molecules, 27(19), 2022. doi:10.3390/molecules27196772.

Bibtex

@article{sacscuza:bilal2024mssrmsessg,
  title={Modified Sombor Spectral Radii and Modified Sombor Energies of Splitting and Shadow Graphs},
  author={A. Bilal, M. Mobeen Munir},
  journal={Scientific Annals of Computer Science},
  volume={34},
  number={2},
  organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania},
  year={2024},
  pages={89-112},
  publisher={Alexandru Ioan Cuza University Press, Ia\c si},
  doi={10.47743/SACS.2024.2.89}
}