Topological Efficiency of Some Product Graphs

Document Type: Research Paper

Authors

Annamalai University

10.22052/ijmc.2017.82177.1280

Abstract

The topological efficiency index of a connected graph $G,$ denoted by $\rho (G),$ is defined as $\rho(G)=\frac{2W(G)}{\left|V(G)\right|\underline w(G)},$ where $\underline w(G)=\text { min }\left\{w_v(G):v\in V(G)\right\}$ and $W(G)$ is the Wiener index of $G.$ In this paper, we obtain the value of topological efficiency index for some composite graphs such as tensor product, strong product, symmetric difference and disjunction of two connected graphs. Further, we have obtained the topological efficiency index for a double graph of a given graph.

Keywords


1. N. Alon and E. Lubetzky, Independent set in tensor graph powers, J. Graph
Theory 54 (2007) 73–87.
2. B. Bresar, W. Imrich, S. Klavžar and B. Zmazek, Hypercubes as direct products,
SIAM J. Discrete. Math. 18 (2005) 778–786.
3. S. Hossein-Zadeh, A. Iranmanesh, M. A. Hossein-Zadeh and A. R. Ashrafi,
Topological efficiency under graph operations, J. Appl. Math. Comput. 54 (2017)
69–80.
4. O. Ivanciuć, QSAR comparative study of Wiener descriptors for weighted
molecular graphs, J. Chem. Inf. Comput. Sci. 40 (2000) 1412–1422.

5. O. Ivanciuć, T. S. Balaban and A. T. Balaban, Reciprocal distance matrix, related
local vertex invariants and topological indices, J. Math. Chem. 12 (1993) 309–
318.
6. W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley,
New York, 2000.
7. F. Koorepazan-Moftakhar, A. R. Ashrafi, O. Ori and M. V. Putz, Topological
efficiency of fullerene, J. Comput. Theor. Nanosci. 12 (2015) 971–975.
8. F. Koorepazan-Moftakhar, A. R. Ashrafi, O. Ori and M. V. Putz, Topological
invariants of nanocones and fullerenes, Curr. Org. Chem. 19 (2015) 240–248.
9. H. Lei, T. Li, Y. Shi and H. Wang, Wiener polarity index and its generalization in
trees, MATCH Commun. Math. Comput. Chem. 78 (2017) 199–212.
10. S. Li and Y. Song, On the sum of all distances in bipartite graphs, Discrete Appl.
Math. 169 (2014) 176–185.
11. S. C. Li and W. Wei, Some edge-grafting transformation on the eccentricity
resistance-distance sum and their applications, Discrete Appl. Math. 211 (2016)
130–142.
12. J. Ma, Y. Shi, Z. Wang and J. Yue, On Wiener polarity index of bicyclic
networks, Sci. Rep. 6 (2016) 19066.
13. K. Pattabiraman and P. Paulraja, On some topological indices of the tensor
products of graphs, Discreate Appl. Math. 160 (2012) 267–79.
14. K. Pattabiraman and P. Paulraja, Wiener and vertex PI indices of the strong
product of graphs, Discuss. Math. Graph Theory 32 (2012) 749–769.
15. S. Sardana and A. K. Madan, Predicting anti-HIV activity of TIBO derivatives: A
computational approach using a novel topological descriptor, J. Mol. Model 8
(2002) 258–265.
16. D. Vukičević, F. Cataldo, O. Ori and A. Graovac, Topological efficiency of C66
fullerene, Chem. Phys. Lett. 501 (2011) 442–445.
17. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc.
69 (1947) 17–20.
18. H. Zhang, S. Li and L. Zhao, On the further relation between the (revised) Szeged
index and the Wiener index of graphs, Discrete Appl. Math. 206 (2016) 152–164.