A study on the temperature Sombor energy and entropy of a graph
Resumen
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.
Descargas
La descarga de datos todavía no está disponible.
Citas
\bibitem{furt15} B. Furtula, I. Gutman, A forgotten topological index., \textit{Journal of Mathematical Chemistry} {\bf 53(4)} (2015), 1184-1190.
\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.
\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.
Publicado
2025-10-09
Sección
Special Issue on “Applied Mathematics and Computing”(ICAMC-25)
Derechos de autor 2025 Boletim da Sociedade Paranaense de Matemática
Esta obra está bajo licencia internacional Creative Commons Reconocimiento 4.0.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
