Opuscula Math. 38, no. 1 (2018), 81-94
https://doi.org/10.7494/OpMath.2018.38.1.81

Opuscula Mathematica

# Wiener index of strong product of graphs

Iztok Peterin
Petra Žigert Pleteršek

Abstract. The Wiener index of a connected graph $$G$$ is the sum of distances between all pairs of vertices of $$G$$. The strong product is one of the four most investigated graph products. In this paper the general formula for the Wiener index of the strong product of connected graphs is given. The formula can be simplified if both factors are graphs with the constant eccentricity. Consequently, closed formulas for the Wiener index of the strong product of a connected graph $$G$$ of constant eccentricity with a cycle are derived.

Keywords: Wiener index, graph product, strong product.

Mathematics Subject Classification: 05C12, 05C76.

Full text (pdf)

1. A.R. Ashrafi, S. Yousefi, Computing the Wiener index of a TUC4C8(S) nanotorus, MATCH Commun. Math. Comput. Chem. 57 (2007), 403-410.
2. R.M. Casablanca, O. Favaron, M. Kouider, Average distance in the strong product of graphs, Utilitas Math. 94 (2014), 31-48.
3. F.G. Chung, The average distance and the independence number, J. Graph Theory 12 (1988), 229-235.
4. P. Dankelmann, Average distance and the independence number, Discrete Appl. Math. 51 (1994), 73-83.
5. P. Dankelmann, Average distance and the domination number, Discrete Appl. Math. 80 (1997), 21-35.
6. P. Dankelmann, Average distance and generalized packing in graphs, Discrete Math. 310 (2010), 2334-2344.
7. A.A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener Index of Hexagonal Systems, Acta Appl. Math. 72 (2002), 247-294.
8. R.C. Entringer, D.E. Jackson, D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976), 283-296.
9. M. Ghorbani, M.A. Hosseinzadeh, On Wiener index of special case of link of fullerenes, Optoelectron Adv. Mat. Journal 4 (2010), 538-539.
10. A. Graovac, T. Pisanski, On the Wiener index of a graph, J. Math. Chem. 8 (1991), 53-62.
11. R. Hammack, W. Imrich, S. Klavžar, Handbook of Product Graphs, Second Edition, CRC Press, Boca Raton, FL, 2011.
12. D.J. Klein, I. Lukovits, I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 35 (1995), 50-52.
13. K. Pattabirman, Exact Wiener indices of the strong product of graphs, J. Prime Res. Math. 9 (2013), 18-33.
14. K. Pattabiraman, P. Paulraja, Wiener index of the tensor product of a path and a cycle, Discuss. Math. Graph Theory 31 (2011), 737-751.
15. K. Pattabiraman, P. Paulraja, On some topological indices of the tensor products of graphs, Discrete Appl. Math. 160 (2012), 267-279.
16. K. Pattabiraman, P. Paulraja, Wiener and vertex PI indices of the strong product of graphs, Discuss. Math. Graph Theory 32 (2012), 749-769.
17. H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947), 17-20.
18. Y.N. Yeh, I. Gutman, On the sum of all distances in composite graphs, Discrete Math. 135 (1994), 359-365.
• Iztok Peterin
• University of Maribor, Faculty of Electrical Engineering and Computer Science, Koroška 46, 2000 Maribor, Slovenia
• Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
• Petra Žigert Pleteršek
• University of Maribor, Faculty of Chemistry and Chemical Engineering, Smetanova 17, 2000 Maribor, Slovenia
• Institute of Mathematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
• Communicated by Mirko Horňák.
• Revised: 2017-05-04.
• Accepted: 2017-05-04.
• Published online: 2017-11-13.