PDF
Discussiones Mathematicae Graph Theory 32(1) (2012)
121-127
DOI: https://doi.org/10.7151/dmgt.1590
p-Wiener Intervals and p-Wiener Free Intervals
| Kumarappan Kathiresan
Center for Research and Post Graduate Studies in Mathematics | S. Arockiaraj
Department of Mathematics |
Abstract
A positive integer n is said to be Wiener graphical, if there exists a graph G with Wiener index n. In this paper, we prove that any positive integer n(≠ 2,5) is Wiener graphical. For any positive integer p, an interval [a,b] is said to be a p-Wiener interval if for each positive integer n ∈ [a,b] there exists a graph G on p vertices such that W(G) = n. For any positive integer p, an interval [a,b] is said to be p-Wiener free interval (p-hyper-Wiener free interval) if there exist no graph G on p vertices with a ≤ W(G) ≤ b (a ≤ WW(G) ≤ b). In this paper, we determine some p-Wiener intervals and p-Wiener free intervals for some fixed positive integer p.
Keywords: Wiener index of a graph, Wiener graphical, p-Wiener interval, p-Wiener free interval, hyper-Wiener index of a graph, radius, diameter
2010 Mathematics Subject Classification: 05C12.
References
| [1] | F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley Reading, 1990). |
| [2] | P.G. Doyle and J.L. Snell, Random Walks and Electric Networks (Math. Assoc., Washington, 1984). |
| [3] | R.C. Entringer, D.E. Jackson and D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (101) 1976. |
| [4] | D. Goldman, S. Istrail, G. Lancia and A. Picolboni, Algorithmic strategies in Combinatorial Chemistry, in: 11th ACM-SIAM Symposium, Discrete Algorithms (2000) 275--284. |
| [5] | I. Gutman, Y.N. Yeh, S.L. Lee and J.C. Chen, Wiener number of dendrimers, Comm. Math. Chem., 30 (1994) 103--115. |
| [6] | I. Gutman, Relation between hyper-Wiener and Wiener index, Chem. Phys. Lett. 364 (2002) 352--356, doi: 10.1016/S0009-2614(02)01343-X. |
| [7] | KM. Kathiresan and S. Arockiaraj, Wiener indices of generalized complementary prisms, Bull. Inst. Combin. Appl. 59 (2010) 31--45. |
| [8] | S. Klavžar, P. Zigert and I. Gutman, An algorithm for the calculation of the hyper-Wiener index of benzenoid hydrocarbons, Comput. Chem. 24 (2000) 229--233, doi: 10.1016/S0097-8485(99)00062-5. |
| [9] | Liu Mu-huo and Xuezhong Tan, The first to (k+1)-th smallest Wiener (Hyper -Wiener) indices of connected graphs, Kragujevac J. Math. 32 (2009) 109--115. |
| [10] | S. Nikolić, N. Trinajstić and Z. Mihalić, The Wiener index: Development and applications, Croat. Chem. Acta. 68 (1995) 105--129. |
| [11] | H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) 17--20, doi: 10.1021/ja01193a005. |
Received 8 July 2010
Revised 15 February 2011
Accepted 15 February 2011
Close