A study on the temperature Sombor energy and entropy of a graph
Résumé
The temperature Sombor index is one of the variations of the recently introduced Sombor index, a degree based topological index, found to have nice mathematical properties and very useful applications. In our current study, we introduce the temperature Sombor matrix $\mathcal{T}(G)$, an associated matrix of the temperature Sombor index of a graph $G$, and present certain bounds on its eigenvalues. Additionally, we define the temperature Sombor energy $\mathcal{ET(G)}$ of $G$ and determine some bounds on it. We also discuss the chemical applicability of this parameter by comparing it with the $\pi$-electron energy of certain organic compounds. Additionally, we perform the regression analysis of the temperature Sombor energy with the graph energy of trees with fixed orders $n=8, 9, \dots, 18$. Further, we compute the temperature Sombor entropy of the silicon carbide compound and analyze it in conjunction with its temperature Sombor energy.
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Références
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\bibitem{citekey} B. Zhou, I. Gutman, T. Aleksic, A note on Laplacian energy of graphs., \textit{MATCH Commun. Math. Comput. Chem} {\bf 60(2)} (2008), 441-446.
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\bibitem{shan48} C. E. Shannon, A mathematical theory of communication., \textit{The Bell System Technical Journal} {\bf 27(3)} (1948), 379-423.
\bibitem{fli20}F. Li, X. Li, H. Broersma, Spectral properties of inverse sum indeg index of graphs., \textit{Journal of Mathematical Chemistry} {\bf 58} (2020), 2108-2139.
\bibitem{fmova23}F. Movahedi, M. H. Akhbari, Entire Sombor index of graphs., \textit{Iranian Journal of Mathematical Chemistry} {\bf 14(1)} (2023), 33-45.
\bibitem{bore21} H. S. Boregowda, R. B. Jummannaver, Neighbors degree sum energy of graphs., \textit{Journal of Applied Mathematics and Computing} {\bf 67(1)} (2021), 579-603.
\bibitem{lee07}H. Lee, Topics in Inequalities-Theorems and Techniques. The IMO Compendium Group (2007).
\bibitem{hli22} H.Liu, Multiplicative Sombor index of graphs., \textit{Discrete Math. Lett} {\bf 9} (2022), 80-85.
\bibitem{gutm72} I. Gutman, N. Trinajsti$\acute{c}$, Graph theory and molecular orbitals. Total $\pi$-electron energy of alternant hydrocarbons., \textit{Chem. Phys. Lett.} {\bf 17} (1972), 535–538.
\bibitem{gutm78} I. Gutman, The energy of a graph., \textit{Ber. Math.— Statist. Sekt. Forsch-ungsz. Graz.} {\bf 103} (1978), 1-22.
\bibitem{gutm94} I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles., \textit{Graph Theory Notes NY} {\bf 27(9)} (1994), 9-15.
\bibitem{gutm211} I. Gutman, Some basic properties of Sombor indices., \textit{Open J. Discrete Appl. Math.} {\bf 4(1)} (2021), 1–3.
\bibitem{gutm23} I. Gutman, Note on the temperature Sombor index., \textit{Vojnotehnicki Glasnik} {\bf71(3)} (2023), 507-515.
\bibitem{gutm24}I. Gutman, B. Furtula, M. S. Oz, Geometric approach to vertex‐degree‐based topological indices–Elliptic Sombor index, theory and application., \textit{International Journal of Quantum Chemistry} {\bf124(2)} (2024), e27346.
\bibitem{diaz63}J. B. Diaz, F. T. Metcalf, Stronger forms of a class of inequalities of G. Polya-G. Szego, and LV Kantorovich., \textit{Bulletin of the American Mathematical Society} {\bf 69(3)} (1963), 415-418.
\bibitem{jrad21} J. Rada, J. M. Rodr$\acute{i}$guez, J. M. Sigarreta, General properties on Sombor indices., \textit{Discrete Appl. Math.} {\bf 299} (2021), 87-97.
\bibitem{viji23} J. S. Vijay, S. Roy, B. C. Beromeo, M. N. Husin, T. Augustine, R. U. Gobithaasan, M. Easuraja, Topological properties and entropy calculations of aluminophosphates., \textit{Mathematics} {\bf 11(11)} (2023), 2443.
\bibitem{das14} K. C. Das, S. Sorgun, K. Xu, On Randic energy of graphs., \textit{MATCH Commun. Math. Comput. Chem} {\bf72(1)} (2014), 227-238.
\bibitem{kcda21} K. C. Das, A. S. Cevik, I. N. Cangul, Y. Shang, On Sombor index., \textit{Symmetry} {\bf 13} (2021), 140.
\bibitem{das16} K. Das, S. A. Mojalal, On energy and Laplacian energy of graphs., \textit{The Electronic Journal of Linear Algebra} {\bf 31} (2016), 167-186.
\bibitem{mm08} K. I. Ramachandran, , G. Deepa, K. Namboodiri, Computational Chemistry and Molecular Modelling: Principles and Applications.,\textit{ Springer}, Berlin (2008).
\bibitem{gowt21} K. J. Gowtham, N. Narahari, On Sombor energy of graphs., \textit{Nanosystems: Physics, Chemistry and Mathematics}, {\bf 12(4)} (2021), 411-417.
\bibitem{Yk78} K. Yates, Huckel Molecular Orbital Theory., Academic Press, New York, 1978.
\bibitem{zhong12} L. Zhong, The harmonic index for graphs., \textit{Applied Mathematics Letters} {\bf 25(3)} (2012), 561-566.
\bibitem{rand75}M.~Randi$\acute{c}$, {On characterization of molecular branching}., \textit{Journal of American Chemical Society} \textbf{97} (1975), 6609-6615.
\bibitem{ani23} M. U. Ani, F. J. H. Campena, S. Ali, S. Dehraj, M. Cancan, F. M. Alharbi, A. M. Galal, Characterizations of chemical networks entropies by K-banhatii topological indices., \textit{Symmetry} {\bf 15(1)} (2023), 143.
\bibitem{hari24} N. Harish, B. Sarveshkumar, B. Chaluvaraju, The reformulated Sombor index of a graph., \textit{Transactions on Combinatorics} {\bf 13(1)} (2024), 1-16.
\bibitem{nara24}N. Narahari, K. J. Gowtham, B. Sooryanarayana, Reverse Sombor Energy of a Graph., \textit{Biointerface Research in Applied Chemistry} {\bf 14(3)} (2024), 61.
\bibitem{nigar23} N. Nigar, S. Alam, M. Rasheed, M. Farahani, M. Alaeiyan, M. Cancan, On Ve-Degree and Ev-Degree Based Topological Invariants of Chemical Structures, \textit{Mathematics} {\bf 119)} (2023), 25-35.
\bibitem{oboud16} Oboudi, Mohammad Reza, Energy and Seidel energy of graphs., \textit{MATCH Commun. Math. Comput. Chem} {\bf75(2)} (2016), 291-303.
\bibitem{aguil21}R. Aguilar-S$\acute{a}$nchez, J. A. M$\acute{e}$ndez-Bermudez, J. M. Rodriguez, J. M. Sigarreta, Normalized Sombor Indices as Complexity Measures of Random Networks., \textit{Entropy} {\bf 23} (2021), 976.
\bibitem{cruz21} R. Cruz, I. Gutman, J. Rada, Sombor index of chemical graphs., \textit{Applied Mathematics and Computation} {\bf 399} (2021), 126018.
\bibitem{cruz21}R. Cruz, J. Rada, J. Sigarreta, Sombor index of trees with at most three branch vertices., \textit{Applied Mathematics and Computation} {\bf 409} (2021), 126414.
\bibitem{haun24} R. Huang, M. F. Hanif, M. K. Siddiqui, M. F. Hanif, M. Hussain, F. B. Petros, Exploring topological indices and entropy measure via rational curve fitting models for calcium hydroxide network., \textit{ Scientific Reports} {\bf 14(1)} (2024), 23452.
\bibitem{alik14} S. Alikhani, N. Ghanbari, More on energy and Randic energy of specific graphs., \textit{ArXiv} (2014).
\bibitem{sberd01} S. Bernard, J. M. Child, Higher Algebra. Macmillan India Ltd, New Delhi (2001).
\bibitem{sfil21} S. Filipovski, Relations between Sombor index and some topological indices., \textit{Iranian Journal of Mathematical Chemistry} {\bf 12} (2021), 19–26.
\bibitem{sfaj88} S. Fajtolowicz, On conjectures of Graffitti., \textit{Discrete Math.} {\bf 72} (1988), 113-118.
\bibitem{shayat24} S. Hayat, A. Khan, K. Ali, J. B. Liu, Structure-property modeling for thermodynamic properties of benzenoid hydrocarbons by temperature-based topological indices., \textit{Ain Shams Engineering Journal} {\bf 15(3)} (2024), 102586.
\bibitem{haya24} S. Hayat, S. J. Alanazi, J. B. Liu, Two novel temperature-based topological indices with strong potential to predict physicochemical properties of polycyclic aromatic hydrocarbons with applications to silicon carbide nanotubes., \textit{Physica Scripta} {\bf 99(5)} (2024), 055027.
\bibitem{negi24} S. Negi, V. K. Bhat, Face Index of Silicon Carbide Structures: An Alternative Approach., \textit{Silicon} {\bf 16(16)} (2024), 5865-5876.
\bibitem{hosa17} S. M. Hosamani, B. B. Kulkarni, R. G. Boli, V. M. Gadag, QSPR analysis of certain graph theocratical matrices and their corresponding energy., \textit{Applied Mathematics and Nonlinear Sciences} {\bf 2(1)} (2017), 131-150.
\bibitem{ssshin24} S. S. Shinde, J. Macha, H. S. Ramane, Bounds for Sombor eigenvalue and energy of a graph in terms of hyper Zagreb and Zagreb indices., \textit{Palestine Journal of Mathematics} {\bf13(1)} (2024), 9-15.
\bibitem{kull211} V. R. Kulli, $\delta-$Sombor index and its exponential for certain nanotubes., \textit{Annals of Pure and Applied Mathematics} {\bf 23} (2021), 37–42.
\bibitem{banku21} V. R. Kulli, On banhatti-sombor indices., \textit{SSRG International Journal of Applied Chemistry} {\bf 8(1)} (2021), 20-25.
\bibitem{vrkulli22}V. R. Kulli, N. Harish, B. Chaluvaraju, I. Gutman, Mathematical properties of KG Sombor index., \textit{Bulletin of International Mathematical Virtual Institute} {\bf 12(2)} (2022), 379-386.
\bibitem{vrk2022} V. R. Kulli, Temperature Sombor and temperature Nirmala indices., \textit{International Journal of Mathematics and Computer Research} {\bf 10(9)} (2022), 2910-2915.
\bibitem{xshi24}X. Shi, R. Cai, J. Ramezani Tousi, A. A. Talebi, Quantitative Structure–Property
Relationship Analysis in Molecular Graphs of Some Anticancer Drugs with Temperature Indices Approach., \textit{Mathematics} {\bf12(13)} (2024), 1953.
\bibitem{chen14} Z. Chen, M. Dehmer, Y. Shi, A note on distance-based graph entropies., \textit{Entropy} {\bf 16(10)} (2014), 5416-5427.
\bibitem{citekey} B. Zhou, I. Gutman, T. Aleksic, A note on Laplacian energy of graphs., \textit{MATCH Commun. Math. Comput. Chem} {\bf 60(2)} (2008), 441-446.
\bibitem{coul65} C. A. Coulson, J. Streitwieser, Dictionary of $\pi$-Electron Calculations., Freeman, San Francisco(1965).
\bibitem{shan48} C. E. Shannon, A mathematical theory of communication., \textit{The Bell System Technical Journal} {\bf 27(3)} (1948), 379-423.
\bibitem{fli20}F. Li, X. Li, H. Broersma, Spectral properties of inverse sum indeg index of graphs., \textit{Journal of Mathematical Chemistry} {\bf 58} (2020), 2108-2139.
\bibitem{fmova23}F. Movahedi, M. H. Akhbari, Entire Sombor index of graphs., \textit{Iranian Journal of Mathematical Chemistry} {\bf 14(1)} (2023), 33-45.
\bibitem{bore21} H. S. Boregowda, R. B. Jummannaver, Neighbors degree sum energy of graphs., \textit{Journal of Applied Mathematics and Computing} {\bf 67(1)} (2021), 579-603.
\bibitem{lee07}H. Lee, Topics in Inequalities-Theorems and Techniques. The IMO Compendium Group (2007).
\bibitem{hli22} H.Liu, Multiplicative Sombor index of graphs., \textit{Discrete Math. Lett} {\bf 9} (2022), 80-85.
\bibitem{gutm72} I. Gutman, N. Trinajsti$\acute{c}$, Graph theory and molecular orbitals. Total $\pi$-electron energy of alternant hydrocarbons., \textit{Chem. Phys. Lett.} {\bf 17} (1972), 535–538.
\bibitem{gutm78} I. Gutman, The energy of a graph., \textit{Ber. Math.— Statist. Sekt. Forsch-ungsz. Graz.} {\bf 103} (1978), 1-22.
\bibitem{gutm94} I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles., \textit{Graph Theory Notes NY} {\bf 27(9)} (1994), 9-15.
\bibitem{gutm211} I. Gutman, Some basic properties of Sombor indices., \textit{Open J. Discrete Appl. Math.} {\bf 4(1)} (2021), 1–3.
\bibitem{gutm23} I. Gutman, Note on the temperature Sombor index., \textit{Vojnotehnicki Glasnik} {\bf71(3)} (2023), 507-515.
\bibitem{gutm24}I. Gutman, B. Furtula, M. S. Oz, Geometric approach to vertex‐degree‐based topological indices–Elliptic Sombor index, theory and application., \textit{International Journal of Quantum Chemistry} {\bf124(2)} (2024), e27346.
\bibitem{diaz63}J. B. Diaz, F. T. Metcalf, Stronger forms of a class of inequalities of G. Polya-G. Szego, and LV Kantorovich., \textit{Bulletin of the American Mathematical Society} {\bf 69(3)} (1963), 415-418.
\bibitem{jrad21} J. Rada, J. M. Rodr$\acute{i}$guez, J. M. Sigarreta, General properties on Sombor indices., \textit{Discrete Appl. Math.} {\bf 299} (2021), 87-97.
\bibitem{viji23} J. S. Vijay, S. Roy, B. C. Beromeo, M. N. Husin, T. Augustine, R. U. Gobithaasan, M. Easuraja, Topological properties and entropy calculations of aluminophosphates., \textit{Mathematics} {\bf 11(11)} (2023), 2443.
\bibitem{das14} K. C. Das, S. Sorgun, K. Xu, On Randic energy of graphs., \textit{MATCH Commun. Math. Comput. Chem} {\bf72(1)} (2014), 227-238.
\bibitem{kcda21} K. C. Das, A. S. Cevik, I. N. Cangul, Y. Shang, On Sombor index., \textit{Symmetry} {\bf 13} (2021), 140.
\bibitem{das16} K. Das, S. A. Mojalal, On energy and Laplacian energy of graphs., \textit{The Electronic Journal of Linear Algebra} {\bf 31} (2016), 167-186.
\bibitem{mm08} K. I. Ramachandran, , G. Deepa, K. Namboodiri, Computational Chemistry and Molecular Modelling: Principles and Applications.,\textit{ Springer}, Berlin (2008).
\bibitem{gowt21} K. J. Gowtham, N. Narahari, On Sombor energy of graphs., \textit{Nanosystems: Physics, Chemistry and Mathematics}, {\bf 12(4)} (2021), 411-417.
\bibitem{Yk78} K. Yates, Huckel Molecular Orbital Theory., Academic Press, New York, 1978.
\bibitem{zhong12} L. Zhong, The harmonic index for graphs., \textit{Applied Mathematics Letters} {\bf 25(3)} (2012), 561-566.
\bibitem{rand75}M.~Randi$\acute{c}$, {On characterization of molecular branching}., \textit{Journal of American Chemical Society} \textbf{97} (1975), 6609-6615.
\bibitem{ani23} M. U. Ani, F. J. H. Campena, S. Ali, S. Dehraj, M. Cancan, F. M. Alharbi, A. M. Galal, Characterizations of chemical networks entropies by K-banhatii topological indices., \textit{Symmetry} {\bf 15(1)} (2023), 143.
\bibitem{hari24} N. Harish, B. Sarveshkumar, B. Chaluvaraju, The reformulated Sombor index of a graph., \textit{Transactions on Combinatorics} {\bf 13(1)} (2024), 1-16.
\bibitem{nara24}N. Narahari, K. J. Gowtham, B. Sooryanarayana, Reverse Sombor Energy of a Graph., \textit{Biointerface Research in Applied Chemistry} {\bf 14(3)} (2024), 61.
\bibitem{nigar23} N. Nigar, S. Alam, M. Rasheed, M. Farahani, M. Alaeiyan, M. Cancan, On Ve-Degree and Ev-Degree Based Topological Invariants of Chemical Structures, \textit{Mathematics} {\bf 119)} (2023), 25-35.
\bibitem{oboud16} Oboudi, Mohammad Reza, Energy and Seidel energy of graphs., \textit{MATCH Commun. Math. Comput. Chem} {\bf75(2)} (2016), 291-303.
\bibitem{aguil21}R. Aguilar-S$\acute{a}$nchez, J. A. M$\acute{e}$ndez-Bermudez, J. M. Rodriguez, J. M. Sigarreta, Normalized Sombor Indices as Complexity Measures of Random Networks., \textit{Entropy} {\bf 23} (2021), 976.
\bibitem{cruz21} R. Cruz, I. Gutman, J. Rada, Sombor index of chemical graphs., \textit{Applied Mathematics and Computation} {\bf 399} (2021), 126018.
\bibitem{cruz21}R. Cruz, J. Rada, J. Sigarreta, Sombor index of trees with at most three branch vertices., \textit{Applied Mathematics and Computation} {\bf 409} (2021), 126414.
\bibitem{haun24} R. Huang, M. F. Hanif, M. K. Siddiqui, M. F. Hanif, M. Hussain, F. B. Petros, Exploring topological indices and entropy measure via rational curve fitting models for calcium hydroxide network., \textit{ Scientific Reports} {\bf 14(1)} (2024), 23452.
\bibitem{alik14} S. Alikhani, N. Ghanbari, More on energy and Randic energy of specific graphs., \textit{ArXiv} (2014).
\bibitem{sberd01} S. Bernard, J. M. Child, Higher Algebra. Macmillan India Ltd, New Delhi (2001).
\bibitem{sfil21} S. Filipovski, Relations between Sombor index and some topological indices., \textit{Iranian Journal of Mathematical Chemistry} {\bf 12} (2021), 19–26.
\bibitem{sfaj88} S. Fajtolowicz, On conjectures of Graffitti., \textit{Discrete Math.} {\bf 72} (1988), 113-118.
\bibitem{shayat24} S. Hayat, A. Khan, K. Ali, J. B. Liu, Structure-property modeling for thermodynamic properties of benzenoid hydrocarbons by temperature-based topological indices., \textit{Ain Shams Engineering Journal} {\bf 15(3)} (2024), 102586.
\bibitem{haya24} S. Hayat, S. J. Alanazi, J. B. Liu, Two novel temperature-based topological indices with strong potential to predict physicochemical properties of polycyclic aromatic hydrocarbons with applications to silicon carbide nanotubes., \textit{Physica Scripta} {\bf 99(5)} (2024), 055027.
\bibitem{negi24} S. Negi, V. K. Bhat, Face Index of Silicon Carbide Structures: An Alternative Approach., \textit{Silicon} {\bf 16(16)} (2024), 5865-5876.
\bibitem{hosa17} S. M. Hosamani, B. B. Kulkarni, R. G. Boli, V. M. Gadag, QSPR analysis of certain graph theocratical matrices and their corresponding energy., \textit{Applied Mathematics and Nonlinear Sciences} {\bf 2(1)} (2017), 131-150.
\bibitem{ssshin24} S. S. Shinde, J. Macha, H. S. Ramane, Bounds for Sombor eigenvalue and energy of a graph in terms of hyper Zagreb and Zagreb indices., \textit{Palestine Journal of Mathematics} {\bf13(1)} (2024), 9-15.
\bibitem{kull211} V. R. Kulli, $\delta-$Sombor index and its exponential for certain nanotubes., \textit{Annals of Pure and Applied Mathematics} {\bf 23} (2021), 37–42.
\bibitem{banku21} V. R. Kulli, On banhatti-sombor indices., \textit{SSRG International Journal of Applied Chemistry} {\bf 8(1)} (2021), 20-25.
\bibitem{vrkulli22}V. R. Kulli, N. Harish, B. Chaluvaraju, I. Gutman, Mathematical properties of KG Sombor index., \textit{Bulletin of International Mathematical Virtual Institute} {\bf 12(2)} (2022), 379-386.
\bibitem{vrk2022} V. R. Kulli, Temperature Sombor and temperature Nirmala indices., \textit{International Journal of Mathematics and Computer Research} {\bf 10(9)} (2022), 2910-2915.
\bibitem{xshi24}X. Shi, R. Cai, J. Ramezani Tousi, A. A. Talebi, Quantitative Structure–Property
Relationship Analysis in Molecular Graphs of Some Anticancer Drugs with Temperature Indices Approach., \textit{Mathematics} {\bf12(13)} (2024), 1953.
\bibitem{chen14} Z. Chen, M. Dehmer, Y. Shi, A note on distance-based graph entropies., \textit{Entropy} {\bf 16(10)} (2014), 5416-5427.
Publiée
2025-10-09
Rubrique
Special Issue on “Applied Mathematics and Computing”(ICAMC-25)
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
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