ORIGINAL_ARTICLE Topological Efficiency of Some Product Graphs 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. https://ijmc.kashanu.ac.ir/article_102017_57332d712f4df9e69475c3fdbbbe8a3c.pdf 2019-09-01T11:23:20 2020-08-07T11:23:20 269 278 10.22052/ijmc.2017.82177.1280 Wiener index topological efficiency index composite graph Kannan Pattabiraman pramank@gmail.com true 1 Annamalai University Annamalai University Annamalai University LEAD_AUTHOR Tholkappian Suganya suganyatpr@gmail.com true 2 Annamalai University Annamalai University Annamalai University AUTHOR 1. N. Alon and E. Lubetzky, Independent set in tensor graph powers, J. Graph 1 Theory 54 (2007) 73–87. 2 2. B. Bresar, W. Imrich, S. Klavžar and B. Zmazek, Hypercubes as direct products, 3 SIAM J. Discrete. Math. 18 (2005) 778–786. 4 3. S. Hossein-Zadeh, A. Iranmanesh, M. A. Hossein-Zadeh and A. R. Ashrafi, 5 Topological efficiency under graph operations, J. Appl. Math. Comput. 54 (2017) 6 69–80. 7 4. O. Ivanciuć, QSAR comparative study of Wiener descriptors for weighted 8 molecular graphs, J. Chem. Inf. Comput. Sci. 40 (2000) 1412–1422. 9 5. O. Ivanciuć, T. S. Balaban and A. T. Balaban, Reciprocal distance matrix, related 10 local vertex invariants and topological indices, J. Math. Chem. 12 (1993) 309– 11 6. W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley, 12 New York, 2000. 13 7. F. Koorepazan-Moftakhar, A. R. Ashrafi, O. Ori and M. V. Putz, Topological 14 efficiency of fullerene, J. Comput. Theor. Nanosci. 12 (2015) 971–975. 15 8. F. Koorepazan-Moftakhar, A. R. Ashrafi, O. Ori and M. V. Putz, Topological 16 invariants of nanocones and fullerenes, Curr. Org. Chem. 19 (2015) 240–248. 17 9. H. Lei, T. Li, Y. Shi and H. Wang, Wiener polarity index and its generalization in 18 trees, MATCH Commun. Math. Comput. Chem. 78 (2017) 199–212. 19 10. S. Li and Y. Song, On the sum of all distances in bipartite graphs, Discrete Appl. 20 Math. 169 (2014) 176–185. 21 11. S. C. Li and W. Wei, Some edge-grafting transformation on the eccentricity 22 resistance-distance sum and their applications, Discrete Appl. Math. 211 (2016) 23 130–142. 24 12. J. Ma, Y. Shi, Z. Wang and J. Yue, On Wiener polarity index of bicyclic 25 networks, Sci. Rep. 6 (2016) 19066. 26 13. K. Pattabiraman and P. Paulraja, On some topological indices of the tensor 27 products of graphs, Discreate Appl. Math. 160 (2012) 267–79. 28 14. K. Pattabiraman and P. Paulraja, Wiener and vertex PI indices of the strong 29 product of graphs, Discuss. Math. Graph Theory 32 (2012) 749–769. 30 15. S. Sardana and A. K. Madan, Predicting anti-HIV activity of TIBO derivatives: A 31 computational approach using a novel topological descriptor, J. Mol. Model 8 32 (2002) 258–265. 33 16. D. Vukičević, F. Cataldo, O. Ori and A. Graovac, Topological efficiency of C66 34 fullerene, Chem. Phys. Lett. 501 (2011) 442–445. 35 17. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 36 69 (1947) 17–20. 37 18. H. Zhang, S. Li and L. Zhao, On the further relation between the (revised) Szeged 38 index and the Wiener index of graphs, Discrete Appl. Math. 206 (2016) 152–164. 39