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2000
Volume 22, Issue 7
  • ISSN: 1570-1794
  • E-ISSN: 1875-6271

Abstract

Aims

To investigate the -eigenvalues and -energy of various types of graphs, including -fold graphs, strong -fold graphs, and extended bipartite double graphs and establish relationships between the -energy of -fold and strong -fold graphs and the -energy of the original graph , we explore the connection between the -energy of extended bipartite double graphs and their ordinary energy and find the graphs that share equienergetic properties with respect to both the ordinary and Harary matrices.

Background

The -eigenvalues of a graph are the eigenvalues of its Harary matrix . The -energy () of a graph, is the sum of the absolute values of its -eigenvalues. Two connected graphs are said to be -equienergetic if they have equal -energies. They are said to -equienergetic if they have equal -energies. Adjacency and Harary matrices have applications in chemistry, such as finding total π-electron energy, quantitative structure-property relationship (QSPR), .

Objectives

We determined the -spectra of -fold graphs, strong -fold graphs and extended bipartite double graphs and established connections between the -energy of different types of graphs and their original graph for investigating the relationship between the -energy of extended bipartite double graphs and their ordinary energy and the graphs that share equienergetic properties with respect to both the adjacency and Harary matrices.

Methods

Spectral algebraic techniques are used to calculate the -eigenvalues and -energy for each type of graph and compare the -energies of different graphs to identify the equienergetic properties and derive relationships between the -energy of extended double cover graphs and their ordinary energy.

Results

We determined the -spectra of -fold graphs, strong -fold graphs and extended bipartite double graphs and established relationships between the -energy of -fold and strong -fold graphs and the -energy of the original graph . Then, we explored the connection between the -energy of extended bipartite double graphs and their ordinary energy and presented graphs demonstrating equienergetic properties concerning both adjacency and Harary matrices.

Conclusion

The study provides insights into the -eigenvalues, -energy and equienergetic properties of various types of graphs. The established relationships and connections contribute to a deeper understanding of graph spectra and energy properties and the findings enhance the theoretical framework for analyzing equienergetic graphs and their spectral properties.

Scope

Possible extensions of this research could include investigating additional types of graphs and exploring further explicit connections between different graph energies and spectral properties.

Harary matrices are distance-based matrices, which can model distances between atoms in molecular structures and could be useful in organic synthesis to predict how molecular structures behave.

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2025-12-11

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References

  1. JanezicD. MilicevicA. NikolicS. TrinajsticN. Graph-theoretical matrices in chemistry.CRC Press201510.1201/b18389
     [Google Scholar]
  2. IvanciucO. BalabanT.S. BalabanA.T. Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices.J. Math. Chem.199312130931810.1007/BF01164642
     [Google Scholar]
  3. XuK. DasK.C. TrinajstićN. The Harary index of a graph.HeidelbergSpringer201510.1007/978‑3‑662‑45843‑3
     [Google Scholar]
  4. GungorA.D. CevikA.S. On the Harary energy and Harary Estrada index of a graph.MATCH Commun. Math. Comput. Chem.201064280296
     [Google Scholar]
  5. Cvetkovi’cD. RowlinsonP. Simi’cS. An introduction to the theory of graph spectra.Cambridge University Press200910.1017/CBO9780511801518
     [Google Scholar]
  6. LiX. ShiY. GutmanI. Graph energy.New YorkSpringer201210.1007/978‑1‑4614‑4220‑2
     [Google Scholar]
  7. RamaneH.S. ParvathaluB. AshokaK. PirzadaS. On families of graphs which are both adjacency equienergetic and distance equienergetic.Indian J. Pure Appl. Math.202455119820910.1007/s13226‑022‑00355‑1
     [Google Scholar]
  8. GutmanI. TrinajstićN. Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons.Chem. Phys. Lett.197217453553810.1016/0009‑2614(72)85099‑1
     [Google Scholar]
  9. RamaneH.S. ParvathaluB. PatilD.D. AshokaK. Graphs equienergetic with their complements.MATCH Commun. Math. Comput. Chem.2019822471480
     [Google Scholar]
  10. RamaneH.S. ParvathaluB. AshokaK. Energy of strong double graphs.J. Anal.20223031033104310.1007/s41478‑022‑00391‑4
     [Google Scholar]
  11. RamaneH.S. AshokaK. ParvathaluB. PatilD. GutmanI. On complementary equienergetic strongly regular graphs.Discrete Math. Lett.202045055
     [Google Scholar]
  12. RamaneH.S. ParvathaluB. AshokaK. An upper bound for difference of energies of a graph and its complement.Examples and Counterexamples2023310010010.1016/j.exco.2023.100100
     [Google Scholar]
  13. RamaneH.S. ParvathaluB. AshokaK. PatilD. On A-energy and S-energy of certain class of graphs. Acta Universitatis Sapientiae.Informatica202113219521910.2478/ausi‑2021‑0009
     [Google Scholar]
  14. RamaneH.S. PatilD. AshokaK. ParvathaluB. Equienergetic graphs using Cartesian product and generalized composition.Sarajevo J. Math.2021171721
     [Google Scholar]
  15. RamaneH.S. ParvathaluB. AshokaK. Energy of extended bipartite double graphs.Match (Mulh.)202287365366010.46793/match.87‑3.653R
     [Google Scholar]
  16. CuiZ. LiuB. On Harary matrix, Harary index and Harary energy.MATCH Commun. Math. Comput. Chem.201268815823
     [Google Scholar]
  17. DasK.C. Maximum eigenvalues of the reciprocal distance matrix.J. Math. Chem.201047212810.1007/s10910‑009‑9529‑1
     [Google Scholar]
  18. DiudeaM.V. IvanciucO. NikolicS. TrinajsticN. Matrices of reciprocal distance, polynomials and derived numbers.MATCH Commun. Math. Comput. Chem.1997354164
     [Google Scholar]
  19. RamaneH.S. AshokaK. Harary energy of complement of line graphs of regular graphs.Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics202069222122610.31801/cfsuasmas.630087
     [Google Scholar]
  20. RamaneH.S. PatilD. AshokaK. ParvathaluB. Harary spectrum of generalized composition of graphs and Harary equienergetic graphs.J. Algebra Relat. Topics201972314510.22124/jart.2020.14263.1161
     [Google Scholar]
  21. ZhouB. TrinajstićN. Maximum eigenvalues of the reciprocal distance matrix and the reverse Wiener matrix.Int. J. Quantum Chem.2008108585886410.1002/qua.21558
     [Google Scholar]
  22. MedinaL. RodríguezJ. TrigoM. New spectral results for Laplacian Harary matrix and the Harary Laplacian-energy-like applying a matrix order reduction.Mathematics2023121210.3390/math12010002
     [Google Scholar]
  23. JahanbaniA. New bounds for the Harary energy and Harary estrada index of graphs.Mati201914051
     [Google Scholar]
  24. MedinaL. TrigoM. Bounds on the Reciprocal distance energy and Reciprocal distance Laplacian energies of a graph.Linear Multilinear Algebra202270163097311810.1080/03081087.2020.1825607
     [Google Scholar]
  25. MedinaL. TrigoM. Upper bounds and lower bounds for the spectral radius of Reciprocal Distance, Reciprocal Distance Laplacian and Reciprocal Distance signless Laplacian matrices.Linear Algebra Appl.202160938641210.1016/j.laa.2020.09.024
     [Google Scholar]
  26. RakshithB.R. ManjunathaB.J. Reciprocal distance energy of complete multipartite graphs.Asian-European Journal of Mathematics2023166235010310.1142/S1793557123501036
     [Google Scholar]
  27. RamaneH.S. ManjalapurV.V. Harary equienergetic graphs.Int. J. Math. Arch.201568186
     [Google Scholar]
  28. RamaneH.S. JummannaverR.B. Harary spectra and Harary energy of line graphs of regular graphs.Gulf Journal of Mathematics201641394610.56947/gjom.v4i1.55
     [Google Scholar]
  29. BaghipurM. GhorbaniM. GanieH.A. ShangY. On the second-largest reciprocal distance signless Laplacian eigenvalue.Mathematics20219551210.3390/math9050512
     [Google Scholar]
  30. MaY. GaoY. ShaoY. Upper and lower bounds for the spectral radius of generalized reciprocal distance matrix of a graph.Mathematics20221015268310.3390/math10152683
     [Google Scholar]
  31. BapatR. PandaS.K. The spectral radius of the reciprocal distance Laplacian matrix of a graph.Bull. Iran. Math. Soc.20184451211121610.1007/s41980‑018‑0084‑z
     [Google Scholar]
  32. PlavšićD. NikolićS. TrinajstićN. MihalićZ. On the Harary index for the characterization of chemical graphs.J. Math. Chem.199312123525010.1007/BF01164638
     [Google Scholar]
  33. UmmerM. PirzadaS. On the reciprocal distance Laplacian spectral radius of graphs.Commun. Comb. Optim.In Press10.22049/cco.2024.29493.2024
     [Google Scholar]
  34. HuangF. LiX. WangS. On graphs with maximum Harary spectral radius.Appl. Math. Comput.201526693794510.1016/j.amc.2015.05.146
     [Google Scholar]
  35. AlhevazA. BaghipurM. AlizadehY. PirzadaS. On eigenvalues of the reciprocal distance signless Laplacian matrix of graphs.Asian-European Journal of Mathematics20211410215017610.1142/S179355712150176X
     [Google Scholar]
  36. MedinaL. TrigoM. Spectral radius of the Harary matrix of the join product of regular graphs.J. Phys. Conf. Ser.20212090101210310.1088/1742‑6596/2090/1/012103
     [Google Scholar]
  37. HayatS. Distance-based graphical indices for predicting thermodynamic properties of benzenoid hydrocarbons with applications.Comput. Mater. Sci.202323011249210.1016/j.commatsci.2023.112492
     [Google Scholar]
  38. IndulalG. Distance equienergetic graphs.Available from: http://www.facweb.iitkgp.ac.in/~rkannan/gma.html (accessed on 23-10-2024)
  39. IndulalG. International Conference on Graph Connections (ICGC)2020Available from: https://icgc2020.wordpress.com/invitedlectures(accessed on 23-10-2024)
    [Google Scholar]
  40. GutmanI. The energy of a graph.Ber. Math. Statist. Sekt. Forsch. Graz.1978103122
     [Google Scholar]
  41. HararyF. Graph Theory.Addison–Wesley196910.21236/AD0705364
     [Google Scholar]
  42. MunariniE. CippoC.P. ScagliolaA. SalviN.Z. Double graphs.Discrete Math.20083082-324225410.1016/j.disc.2006.11.038
     [Google Scholar]
  43. MarinoM.C. SalviN.Z. Generalizing double graphs. Atti della Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche.Matematiche e Naturali20078521910.1478/C1A0702002
     [Google Scholar]
  44. VaidyaS.K. PopatK.M. ENERGY OF m-SPLITTING AND m-SHADOW GRAPHS.Far East Journal of Mathematical Sciences (FJMS)201710281571157810.17654/MS102081571
     [Google Scholar]
  45. GanieH.A. PirzadaS. BaskoroE.T. On energy, Laplacian energy and p-fold graphs.Electronic Journal of Graph Theory and Applications2015319410710.5614/ejgta.2015.3.1.10
     [Google Scholar]
  46. AlonN. Eigenvalues and expanders.Combinatorica198662839610.1007/BF02579166
     [Google Scholar]
  47. GursoyA. ÜlkerA. GursoyN.K. On the Harary index of.Int. J. Maps Math.202472122137
     [Google Scholar]
  48. ParvathaluB. RamaneH.S. On diameter and distance energy of complements of regular graphs.SSRN202410.2139/ssrn.4726841
     [Google Scholar]
  49. HornR.A. JohnsonC.R. Matrix analysis.Cambridge University Press201210.1017/CBO9781139020411
     [Google Scholar]
  50. DavisP.J. Circulant MatricesWiley: New York1979
     [Google Scholar]
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Harary Spectra and Energy of Certain Classes of Graphs
Curr. Org. Synth. 22, 791 (2025); https://doi.org/10.2174/0115701794330372241114102237
/content/journals/cos/10.2174/0115701794330372241114102237
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  • Article Type:     Research Article
Keyword(s): equienergetic graphs; harary eigenvalues; harary energy; harary equienergetic graphs; Harary matrix; ordinary energy

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