Opuscula Mathematica
Opuscula Math. 36, no. 1 (2016), 5-23

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)
References ▼
  1. [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. [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. [3] M. Azari, Sharp lower bounds on the Narumi-Katayama index of graph operations, Appl. Math. Comput. 239 (2014), 409-421.
  4. [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. [5] M. Azari, A. Iranmanesh, Computation of the edge Wiener indices of the sum of graphs, Ars Combin. 100 (2011), 113-128.
  6. [6] M. Azari, A. Iranmanesh, Computing the eccentric-distance sum for graph operations, Discrete Appl. Math. 161 (2013) 18, 2827-2840.
  7. [7] F. Cataldo, O. Ori, A. Graovac, Wiener index of 1-pentagon fullerenic infinite lattice, Int. J. Chem. Model. 2 (2010), 165-180.
  8. [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. [9] M.V. Diudea, QSPR/QSAR studies by molecular descriptors, NOVA, New York, 2001.
  10. [10] M.V. Diudea, Wiener index of dendrimers, MATCH Commun. Math. Comput. Chem. 32 (1995), 71-83.
  11. [11] T. Došlić, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp. 1 (2008), 66-80.
  12. [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. [13] I. Gutman, An exceptional property of first Zagreb index, MATCH Commun. Math. Comput. Chem. 72 (2014), 733-740.
  14. [14] I. Gutman, A property of the Wiener number and its modifications, Indian J. Chem. 36 (1997) A, 128-132.
  15. [15] I. Gutman, Degree-based topological indices, Croat. Chem. Acta. 86 (2013), 351-361.
  16. [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. [17] H. Hosoya, On some counting polynomials in chemistry, Discrete Appl. Math. 19 (1988), 239-257.
  18. [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. [19] D.J. Klein, T. Došlić, D. Bonchev, Vertex-weightings for distance-moments and thorny graphs, Discrete Appl. Math. 155 (2007), 2294-2302.
  20. [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. [21] T. Réti, On the relationships between the first and second Zagreb indices, MATCH Commun. Math. Comput. Chem. 68 (2012), 169-188.
  22. [22] T. Réti, I. Gutman, Relations between ordinary and multiplicative Zagreb indices, Bull. Inter. Math. Virtual Inst. 2 (2012), 133-140.
  23. [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. [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. [25] D. Stevanović, Hosoya polynomials of composite graphs, Discrete Math. 235 (2001), 237-244.
  26. [26] N. Trinajstić, Chemical graph theory, CRC Press, Boca Raton, FL, 1992.
  27. [27] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), 17-20.
  28. [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.
About the authors ▼
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
About the article ▼
Communicated by Dalibor Fronček.
Received: 2014-12-18.
Revised: 2015-06-14.
Accepted: 2015-06-16.
Published online: 2015-09-19.
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: