Opuscula Math. 36, no. 1 (2016), 5-23

Opuscula Mathematica

Vertex-weighted Wiener polynomials of subdivision-related graphs

Mahdieh Azari
Ali Iranmanesh
Tomislav Došlić

Abstract. Singly and doubly vertex-weighted Wiener polynomials are generalizations of both vertex-weighted Wiener numbers and the ordinary Wiener polynomial. In this paper, we show how the vertex-weighted Wiener polynomials of a graph change with subdivision operators, and apply our results to obtain vertex-weighted Wiener numbers.

Keywords: vertex-weighted Wiener numbers, vertex-weighted Wiener polynomials, subdivision graphs.

Mathematics Subject Classification: 05C76, 05C12, 05C07.

Full text (pdf)

  1. Y. Alizadeh, A. Iranmanesh, T. Došlić, M. Azari, The edge Wiener index of suspensions, bottlenecks, and thorny graphs, Glas. Mat. Ser. III 49 (2014) 1, 1-12.
  2. V. Andova, N. Cohen, R. Škrekovski, A note on Zagreb indices inequality for trees and unicyclic graphs, Ars Math. Contemp. 5 (2012) 1, 73-76.
  3. M. Azari, Sharp lower bounds on the Narumi-Katayama index of graph operations, Appl. Math. Comput. 239 (2014), 409-421.
  4. M. Azari, A. Iranmanesh, Chemical graphs constructed from rooted product and their Zagreb indices, MATCH Commun. Math. Comput. Chem. 70 (2013) 3, 901-919.
  5. M. Azari, A. Iranmanesh, Computation of the edge Wiener indices of the sum of graphs, Ars Combin. 100 (2011), 113-128.
  6. M. Azari, A. Iranmanesh, Computing the eccentric-distance sum for graph operations, Discrete Appl. Math. 161 (2013) 18, 2827-2840.
  7. F. Cataldo, O. Ori, A. Graovac, Wiener index of 1-pentagon fullerenic infinite lattice, Int. J. Chem. Model. 2 (2010), 165-180.
  8. M.R. Darafsheh, M.H. Khalifeh, Calculation of the Wiener, Szeged, and PI indices of a certain nanostar dendrimer, Ars Combin. 100 (2011), 289-298.
  9. M.V. Diudea, QSPR/QSAR studies by molecular descriptors, NOVA, New York, 2001.
  10. M.V. Diudea, Wiener index of dendrimers, MATCH Commun. Math. Comput. Chem. 32 (1995), 71-83.
  11. T. Došlić, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp. 1 (2008), 66-80.
  12. B. Furtula, I. Gutman, H. Lin, More trees with all degrees odd having extremal Wiener index, MATCH Commun. Math. Comput. Chem. 70 (2013), 293-296.
  13. I. Gutman, An exceptional property of first Zagreb index, MATCH Commun. Math. Comput. Chem. 72 (2014), 733-740.
  14. I. Gutman, A property of the Wiener number and its modifications, Indian J. Chem. 36 (1997) A, 128-132.
  15. I. Gutman, Degree-based topological indices, Croat. Chem. Acta. 86 (2013), 351-361.
  16. I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total \(\pi\)-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535-538.
  17. H. Hosoya, On some counting polynomials in chemistry, Discrete Appl. Math. 19 (1988), 239-257.
  18. M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, The hyper-Wiener index of graph operations, Comput. Math. Appl. 56 (2008) 5, 1402-1407.
  19. D.J. Klein, T. Došlić, D. Bonchev, Vertex-weightings for distance-moments and thorny graphs, Discrete Appl. Math. 155 (2007), 2294-2302.
  20. R. Nasiri, H. Yousefi-Azari, M.R. Darafsheh, A.R. Ashrafi, Remarks on the Wiener index of unicyclic graphs, J. Appl. Math. Comput. 41 (2013), 49-59.
  21. T. Réti, On the relationships between the first and second Zagreb indices, MATCH Commun. Math. Comput. Chem. 68 (2012), 169-188.
  22. T. Réti, I. Gutman, Relations between ordinary and multiplicative Zagreb indices, Bull. Inter. Math. Virtual Inst. 2 (2012), 133-140.
  23. B.E. Sagan, Y.N. Yeh, P. Zhang, The Wiener polynomial of a graph, Inter. J. Quantum Chem. 60 (1996) 5, 959-969.
  24. J. Sedlar, D. Vukičević, F. Cataldo, O. Ori, A. Graovac, Compression ratio of Wiener index in 2D-rectangular and polygonal lattices, Ars Math. Contemp. 7 (2014) 1, 1-12.
  25. D. Stevanović, Hosoya polynomials of composite graphs, Discrete Math. 235 (2001), 237-244.
  26. N. Trinajstić, Chemical graph theory, CRC Press, Boca Raton, FL, 1992.
  27. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), 17-20.
  28. W. Yan, B.Y. Yang, Y.N. Yeh, The behavior of Wiener indices and polynomials of graphs under five graph decorations, Appl. Math. Lett. 20 (2007), 290-295.
  • Mahdieh Azari
  • Department of Mathematics Kazerun Branch, Islamic Azad University, P.O. Box: 73135-168, Kazerun, Iran
  • Ali Iranmanesh
  • Tarbiat Modares University, Faculty of Mathematical Sciences, Department of Pure Mathematics, P.O. Box: 14115-137, Tehran, Iran
  • Tomislav Došlić
  • University of Zagreb, Faculty of Civil Engineering, Kaciceva 26, 10000 Zagreb, Croatia
  • Communicated by Dalibor Fronček.
  • Received: 2014-12-18.
  • Revised: 2015-06-14.
  • Accepted: 2015-06-16.
  • Published online: 2015-09-19.
Opuscula Mathematica - cover

Cite this article as:
Mahdieh Azari, Ali Iranmanesh, Tomislav Došlić, Vertex-weighted Wiener polynomials of subdivision-related graphs, Opuscula Math. 36, no. 1 (2016), 5-23, http://dx.doi.org/10.7494/OpMath.2016.36.1.5

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.