Distance-based topological indices of tensor product of graphs

Document Type: Research Paper

Authors

University of Kashan

Abstract

Let G and H be connected graphs. The tensor product G + H is a graph with vertex set V(G+H) = V (G) X V(H) and edge set E(G + H) ={(a , b)(x , y)| ax ∈ E(G) & by ∈ E(H)}. The graph H is called the strongly triangular if for every vertex u and v there exists a vertex w adjacent to both of them. In this article the tensor product of G + H under some distancebased topological indices are investigated, when H is a strongly triangular graph. As a special case most of results given by Hoji, Luob and Vumara in [Wiener and vertex PI indices of Kronecker products of graphs, Discrete Appl. Math., 158 (2010), 1848-1855] will be deduced.

Keywords


A. T. Balaban, D. Mills, O. Ivanciuc and S. C. Basak, Reverse Wiener indices,
Croat. Chem. Acta 73 (2000) 923–941.
2. A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and
applications, Acta Appl. Math. 66 (2001), 211–249.
3. A. A. Dobrynin, I. Gutman, S. Klavžar and P. Žigert, Wiener index of hexagonal
systems, Acta Appl. Math. 72 (2002), 247–294.
4. G. H. FathTabar, A. Hamzeh and S. HosseinZadeh, GA2 index of some graph
operations, FILOMAT 24 (2010), 21–28.
5. B. Furtula, I. Gutman, Z. Tomovic, A. Vesel and I. Pesek, Wiener-type topological
indices of phenylenes, Indian J. Chem. 41A (2002) 1767–1772.
6. I. Gutman, A property of the Wiener number and its modifications, Indian J. Chem.
36A (1997) 128–132.
7. I. Gutman, A. A. Dobrynin, S. Klavžar, L. Pavlovic, Wienertype invariants of trees
and their relation, Bull. Inst. Combin. Appl. 40 (2004) 23–30.
8. 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.
9. M. Hoji, Z. Luob and E. Vumara, Wiener and vertex PI indices of Kronecker
products of graphs, Discrete Appl. Math. 158 (2010) 1848–1855.
10. H. Hosoya, Topological index, a newly proposed quantity characterizing the
topological nature of structure isomers of saturated hydrocarbons, Bull. Chem. Soc.
Jpn. 44 (1971) 2332–2339.
11. S. HosseinZadeh, A. Hamzeh and A. R. Ashrafi, Wienertype invariants of some
graph operations, FILOMAT 23(2009) 103–113.
12. S. Klavžar and I. Gutman, A theorem on Wienertype invariants for isometric
subgraphsofhypercubes, Appl. Math. Lett. 19 (2006) 1129–1133.
13. M. H. Khalifeh, H. YousefiAzari, A. R. Ashrafi and I. Gutman, The Edge Szeged
Index of Product Graphs, Croat. Chem. Acta 81 (2) (2008) 277–281.
14. M. H. Khalifeh, H. YousefiAzari and A. R. Ashrafi, The hyper-Wiener index of
graph operations, Comput. Math. Appl. 56 (2008) 1402–1407.
15. M. H. Khalifeh, H. YousefiAzari, A. R. Ashrafi and S.G. Wagner, Some new
results on distancebased graph invariants, Eur. J. Combin. 30 (2009) 1149–1163.
16. M. Khalifeh, H. YousefiAzari and A. R. Ashrafi, The first and second Zagreb
indices of some graph operations, Discrete Appl. Math. 157 (2009) 804–811.
17. D. J. Klein, I. Lukovits and I. Gutman, On the definition of the hyper-Wiener index
for cycle-containing structures, J. Chem. Inf. Comput. Sci. 35 (1995) 50–52.
18. W. Imrich, S. Klavžar, Product Graphs: Structure and Recognition, Wiley, 2000.
19. W. Luo, B. Zhou, Further properties of reverse Wiener index, MATCH. Commun.
Math. Comput. Chem. 61 (2009) 653–661.
20. M. Metsidik, W. Zhang and F. Duan, Hyper and reverseWiener indices of Fsums
of graphs, Discrete Appl. Math. 158 (2010) 1433–1440.
21. H. Wiener, Structural determination of the parafin boiling points, J. Am. Chem. Soc.
69 (1947) 17–20.
22. H. YousefiAzari, B. Manoochehrian, A. R. Ashrafi, The PI index of product
graphs, Appl. Math. Lett. 21 (2008) 624–627.
23. H. Y. Zhu, D. J. Klein and I. Lukovits, Extensions of the Wiener number, J. Chem.
Inf. Comput. Sci. 36 (1996) 420–428.