On Reciprocal Complementary Wiener Index of a Graph

Document Type : Research Paper


Karnatak University, Dharwad


The eccentricity of a vertex v of graph G is the largest distance between and any other vertex of a graph . The reciprocal complementary Wiener (RCW) index of is defined as,
where D is the diameter of G and is the distance between the vertices and . In this paper we have obtained bounds for the index in terms of eccentricities and given an algorithm to compute the index.


Main Subjects

1. F. Buckley, F. Harary, Distances in Graphs, Addison-Wesley, Redwood, 1990.
2. X. Cai, B. Zhou, Reciprocal Complementary Wiener number of trees, unicyclic
graphs and bicyclic graphs, Discrete Appl. Math. 157 (2009) 3046−3054.
3. O. Ivanciuc, QSAR Comparative study of Wiener descriptors for weighted
molecular graphs, J. Chem. Inf. Comput. Sci. 40 (2000) 1412−1422.
4. O. Ivanciuc, T. Ivanciuc, A. T. Balaban, Quantitative structure property relationship
evaluation of structural descriptors derived from the distance and reverse Wiener
matrices, Internet Electron. J. Mol. Des. 1 (2002) 467−487.
5. O. Ivanciuc, T. Ivanciuc, A. T. Balaban, Vertex and edge-weighted molecular
graphs and derived structural descriptors, in: J. Devillers, A. T. Balaban (eds.),
Topological Indices and Related Descriptors in QSAR and QSPR, Gordon and
Breach, Amsterdam, (1999) 169−220.
6. X. Qi, B. Zhou, Extremal properties of reciprocal complementary Wiener number
of trees, Comput. Math. Appl. 62 (2011) 523−531.
7. H. S. Ramane, A. B. Ganagi, H. B. Walikar, Wiener index of graphs in terms of
eccentricities, Iranian J. Math. Chem. 4 (2013) 239−248.
8. H. S. Ramane, V. V. Manjalapur, Reciprocal Wiener index and reciprocal
complementary Wiener index of line graphs, Indian J. Discrete Math. 1 (2015)
9. H. S. Ramane, V. V. Manjalapur, Some bounds for Harary index of graphs, Int. J.
Sci. Engg. Res. 7 (2016) 26−31.
10. N. Trinajstić, Chemical Graph Theory, 2nd revised ed., CRC Press. Boca Raton,
11. H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69
(1947) 17−20.
12. K. Xu, M. Liu, K. C. Das, I. Gutman, B. Furtula, A survey on graphs extremal with
respect distance based topological indices, MATCH Commun. Math. Comput.
Chem. 71 (2004) 461−508.
13. B. Zhou, X. Cai, N. Trinajstić, On reciprocal complementary Wiener number,
Discrete Appl. Math. 157 (2009) 1628−1633.
14. Y. Zhu, F. Wei, F. Li, Reciprocal complementary Wiener numbers of noncaterpillars,
Appl. Math. 7 (2016) 219−226.