On Graphs with Same Distance Distribution

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DOI: 10.4236/am.2017.86062    189 Downloads   290 Views  
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In the present paper we investigate the relationship between Wiener number W, hyper-Wiener number R, Wiener vectors WV, hyper-Wiener vectors HWV, Wiener polynomial H, hyper-Wiener polynomial HH and distance distribution DD of a (molecular) graph. It is shown that for connected graphs G and G*, the following five statements are equivalent: 

; and if G and G* have same distance distribution DD then they have same W and R but the contrary is not true. Therefore, we further investigate the graphs with same distance distribution. Some construction methods for finding graphs with same distance distribution are given.

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Qiu, X. and Guo, X. (2017) On Graphs with Same Distance Distribution. Applied Mathematics, 8, 799-807. doi: 10.4236/am.2017.86062.


[1] Wiener, H. (1947) Structural Determination of Paraffin Boiling Points. Journal of the American Chemical Society, 69, 17-20.
[2] Randic, M. (1993) Novel Molecular Descriptor for Structure-Property Studies. Chemical Physics Letters, 211, 478-483.
[3] Randic, M., Guo, X.F., Oxley, T. and Krishnapriyan, H. (1993) Wiener Matrix: Source of Novel Graph Invariants. Journal of Chemical Information and Computer Sciences, 33, 709-716.
[4] Randic, M., Guo, X.F., Oxley, T., Krishnapriyan, h. and Naylor, L. (1994) Wiener Matrix Invariants. Journal of Chemical Information and Computer Sciences, 34, 361-367.
[5] Klein, D.J. and Gutman, I. (1995) On the Definition of the Hyper-Wiener Index for Cycle-Containing Structures. Journal of Chemical Information and Computer Sciences, 35, 50-52.
[6] Lukovits, I. and Linert, W. (1995) A Novel Definition of the Hyper-Wiener Index for Cycles. Journal of Chemical Information and Computer Sciences, 34, 899-902.
[7] Klein, D.J. and Randic, M. (1993) Resistance Distance. Journal of Mathematical Chemistry, 12, 81-95.
[8] Li, X.H. (2003) The Extended Hyper-Wiener Index. Canadian Journal of Chemistry, 81, 992-996.
[9] Klavzar, S., Gutman, I. and Mohar, B. (1995) Labeling of Benzenoid Systems which Reflects the Vertex-Distance Relations. Journal of Chemical Information and Computer Sciences, 35, 590-593.
[10] Cash, G.G., Klavzar, S. and Petkovsek, M. (2002) Three Methods for Calculation of the Hyper-Wiener Index of Molecular Graphs. Journal of Chemical Information and Computer Sciences, 42, 571-576.
[11] Gutman, I. (2002) Relation between Hyper-Wiener and Wiener Index. Chemical Physics Letters, 364, 352-356.
[12] Hosoya, H. (1998) On Some Counting Polynominals in Chemistry. Discrete Applied Mathematics, 19, 239-257.
[13] Cash, G.G. (2002) Relationship between the Hosoya Polynominal and the Hyper-Wiener Index. Applied Mathematics Letters, 15, 893-895.
[14] Buckley, F. and Superville, L. (1981) Distance Distributions and Mean Distance Problem. Proceedings of Third Caribbean Conference on Combinatorics and Computing, University of the West Indies, Barbados, January 1981, 67-76.
[15] Guo, X.F., klein, D.J., Yan, W.G. and Yeh, Y.-N. (2006) Hyper-Wiener Vector, Wiener Matrix Sequence, and Wiener Polynominal Sequence of a Graph. International Journal of Quantum Chemistry, 106, 1756-1761.

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