Degree Distance and Gutman Index of Two Graph Products

  • Shaban Sedghi
  • Nabi Shobe
Keywords: Degree distance, Adjacency matrix, Distance matrix, Complete product, Strong product


The degree distance was introduced by Dobrynin, Kochetova and Gutman as a weighted version of the Wiener index. In this paper, we investigate the degree distance and Gutman index of complete, and strong product graphs by using the adjacency and distance matrices of a graph.


[1] A. Alwardi, B. Arsic, I. Gutman, N. D. Soner, The common neighborhood graph and its energy, Iran. J. Math. Sci. Inf. 7 (2012) 1–8.
[2] J. A. Bondy, U. S. R. Murty, Graph theory, Springer, New York, 2008.
[3] A. S. Bonifácio, R. R. Rosa, I. Gutman, N. M. M. de Abreu, Complete common neighborhood graphs, Proceedings of Congreso Latino-Iberoamericano de Investigaci on Operativa and Simposio Brasileiro de Pesquisa Operacional (2012) 4026–4032.
[4] S. Chen, Cacti with the smallest, second smallest, and third smallest Gutman index, J. Combin. Optim. 31(1) (2016) 327–332.
[5] S. Chen, Z. Guo, A lower bound on the degree distance in a tree, Int. J. Contemp. Math. Sci. 5(13) (2010) 649–652.
[6] P. Dankelmann, I. Gutman, S. Mukwembi, H.C. Swart, On the degree distance of a graph, Discrete Appl. Math. 157(13) (2009) 2773–2777.
[7] P. Dankelmann, I. Gutman, S. Mukwembi, H. C. Swart, The edge–Wiener index of a graph, Discrete Math. 309 (2009) 3452–457.
[8] A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001) 211–249.
[9] A. A. Dobrynin, A. A. Kochetova, Degree distance of a graph: A degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. 34(5) (1994) 1082–1086.
[10] T. Došlic, B. Furtula, A. Graovac, I. Gutman, S. Moradi, Z. Yarahmadi, On vertex–degree–based molecular structure descriptors, MATCH Commun. Math. Comput. Chem. 66 (2011) 613–626.
[11] A. A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002) 247–294.
[12] L. Feng, W. Liu, The maximal Gutman index of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 66 (2011) 699–708.
[13] I. Gutman, Y. N. Yeh, S. L. Lee, Y. L. Luo, Some recent results in the theory of the Wiener number, Indian J. Chem. 32A (1993) 651–661.
[14] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34(5) (1994) 1087–1089.
[15] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17(4) (1972) 535–538.
[16] P. Paulraja, V.S. Agnes, Degree distance of product graphs, Discrete Math., Alg. and Appl. 6(1) (2014) 1450003.
[17] P. Paulraja, V. S. Agnes, Gutman index of product graphs, Discrete Math., Alg. and Appl. 6(4) (2014) 1450058.
[18] R. Hammack, W. Imrich, Sandi Klav˘zr, Handbook of product graphs, Second edition, CRC Press, 2011.
[19] S. Nikolic, N. Trinajstic, Z. Mihalic, The Wiener index: Development and applications, Croat. Chem. Acta 68 (1995) 105–129.
[20] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20.
[21] H.P. Schultz, Topological organic chemistry. 1. Graph theory and topological indices of alkanes, J. Chem. Inf. Comput. Sci. 29(3) (1989) 227–228.