• Received : 2015.03.13
  • Accepted : 2016.02.16
  • Published : 2016.05.30


In this paper, the exact formulae for the generalized product degree distance, reciprocal product degree distance and product degree distance of tensor product of a connected graph and the complete multipartite graph with partite sets of sizes m0, m1, ⋯ , mr−1 are obtained.


generalized product degree distance;tensor product


  1. A.R. Ashrafi, T. Doslic and A. Hamzeha, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010), 1571-1578.
  2. N. Alon and E. Lubetzky, Independent set in tensor graph powers, J. Graph Theory 54 (2007), 73-87.
  3. A.A. Dobrynin and A.A. Kochetova, Degree distance of a graph: a degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. 34 (1994), 1082-1086.
  4. Y. Alizadeh, A. Iranmanesh and T. Doslic, Additively weighted Harary index of some composite graphs, Discrete Math. 313 (2013), 26-34.
  5. A.M. Assaf, Modified group divisible designs, Ars Combin. 29 (1990), 13-20.
  6. B. Bresar, W. Imrich, S. Klavẑar and B. Zmazek, Hypercubes as direct products, SIAM J. Discrete Math. 18 (2005), 778-786.
  7. S. Chen and W. Liu , Extremal modified Schultz index of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 64 (2010), 767-782.
  8. J. Devillers and A.T. Balaban, Eds., Topological indices and related descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, The Netherlands, 1999.
  9. M.V. Diudea(Ed.), QSPR/QRAR Studies by molecular descriptors, Nova, Huntington, 2001.
  10. B. Furtula, I.Gutman, Z. Tomovic, A. Vesel and I. Pesek, Wiener-type topological indices of phenylenes, Indian J. Chem. 41A (2002), 1767-1772.
  11. I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34 (1994), 1087-1089.
  12. I. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin, 1986.
  13. I. Gutman, A property of the Wiener number and its modifications, Indian J. Chem. 36A (1997), 128-132.
  14. I. Gutman, A.A. Dobrynin, S. Klavzar and L. Pavlovic, Wiener-type invariants of trees and their relation, Bull. Inst. Combin. Appl. 40 (2004), 23-30.
  15. H. Hua and S. Zhang, On the reciprocal degree distance of graphs, Discrete Appl. Math. 160 (2012), 1152-1163.
  16. I. Gutman, D.Vidovic and L. Popovic, Graph representation of organic molecules. Cayley's plerograms vs. his kenograms, J.Chem. Soc. Faraday Trans. 94 (1998), 857-860.
  17. A. Hamzeh, A. Iranmanesh, S. Hossein-Zadeh and M.V. Diudea, Generalized degree distance of trees, unicyclic and bicyclic graphs, Studia Ubb Chemia, LVII 4 (2012), 73-85.
  18. A. Hamzeh, A. Iranmanesh and S. Hossein-Zadeh, Some results on generalized degree distance, Open J. Discrete Math. 3 (2013), 143-150.
  19. W. Imrich and S. Klavẑar, Product graphs: Structure and Recognition, John Wiley, New York, 2000.
  20. S.C. Li and X. Meng, Four edge-grafting theorems on the reciprocal degree distance of graphs and their applications, J. Comb. Optim. 30 (2015), 468-488.
  21. A. Mamut and E. Vumar, Vertex vulnerability parameters of Kronecker products of complete graphs, Inform. Process. Lett. 106 (2008), 258-262.
  22. K. Pattabiraman and M. Vijayaragavan, Reciprocal degree distance of some graph operations, Trans. Comb. 2 (2013), 13-24.
  23. K. Pattabiraman and M. Vijayaragavan, Reciprocal degree distance of product graphs, Accepted in Discrete Appl. Math. 179 (2014), 201-213.
  24. G.F. Su, L.M. Xiong, X.F. Su and X.L. Chen, Some results on the reciprocal sum-degree distance of graphs, J. Comb. Optim., 30 (2015), 435-446.
  25. G. Su, I. Gutman, L. Xiong and L. Xu,Reciprocal product degree distance of graphs, Manuscript.
  26. H.Y. Zhu, D.J. Klenin and I. Lukovits, Extensions of the Wiener number, J. Chem. Inf. Comput. Sci. 36 (1996), 420-428.