The Reciprocal Complementary Wiener Number of Graph Operations


Download PDF

Authors: R. NASIRI, A. NAKHAEI AND A. R. SHOJAEIFARD

DOI: 10.46793/KgJMat2101.139N

Abstract:

The reciprocal complementary Wiener number of a connected graph G is defined as {x,y}⊆V (G)      1
D+1-−-dG(x,y), where D is the diameter of G and dG(x,y) is the distance between vertices x and y. In this work, we study the reciprocal complementary Wiener number of various graph operations such as join, Cartesian product, composition, strong product, disjunction, symmetric difference, corona product, splice and link of graphs.

Keywords:

Reciprocal complementary Wiener number, distance, graph operations.

References:

[1]   F. M. Brückler, T. Došlić, A. Graovac and I. Gutman, On a class of distance-based molecular structure descriptors, Chemical Physics Letters 503 (2011), 336–338.

[2]   X. Cai and B. Zhou, Reciprocal complementary Wiener numbers of trees, unicyclic graphs and bicyclic graphs, Discrete Appl. Math. 157 (2009), 3046–3054.

[3]   A. A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta. Appl. Math. 66 (2001), 211–249.

[4]   W. Imrich and S. Klavzar, Product Graphs: Structure and Recognition, Wiley, New York, 2000.

[5]   O. Ivanciuc, QSAR comparative study of Wiener descriptors for weighted molecular graphs, Journal of Chemical Information and Computer Sciences 40 (2000), 1412–1422.

[6]   O. Ivanciuc, T. Ivanciuc and A. T. Balaban, The complementary distance matrix, a new molecular graph metric, Acta Chimica Hungarica - Models in Chemistry 137 (2000), 57–82.

[7]   D. Janežič, A. Milićević, S. Nikolić and N. Trinajstić, Graph Theoretical Matrices in Chemistry, University of Kragujevac, Kragujevac, 2007.

[8]   M. H. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, The hyper-Wiener index of graph operations, Comput. Math. Appl. 56 (2008), 1402–1407.

[9]   R. Nasiri, H. Yousefi-Azari, M. R. Darafsheh and A. R. Ashrafi, Remarks on the Wiener index of unicyclic graphs, J. Appl. Math. Comput. 41 (2013), 49–59.

[10]   H. Ramane and V. Manjalapur, Reciprocal Wiener index and reciprocal complementary Wiener index of line graphs, Indian Journal of Discrete Mathematics 1 (2015), 23–32.

[11]   Z. K. Tang and H. Y. Deng, The graphs with minimal and maximal Wiener indice among a class of bicyclic graphs, J. Nat. Sci. Hunan Norm. Univ. 31 (2008), 27–30.

[12]   H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947), 17–20.

[13]   K. Xu, M. Liu, K. C. Das, I. Gutman and B. Furtula, A survey on graphs extremal with respect to distance-based topological indices, MATCH Commun. Math. Comput. Chem. 71 (2014), 461–508.

[14]   B. Zhou, X. Cai and N. Trinajstić, On reciprocal complementary Wiener number, Discrete Appl. Math. 157 (2009), 1628–1633.

[15]   Y. Zhu, F. Wei and F. Li, Reciprocal complementary Wiener numbers of non-caterpillars, Appl. Math. 7 (2016), 219–226.