p-Wiener intervals and p-Wiener free intervals

Kumarappan Kathiresan; S. Arockiaraj

Discussiones Mathematicae Graph Theory (2012)

  • Volume: 32, Issue: 1, page 121-127
  • ISSN: 2083-5892

Abstract

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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.

How to cite

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Kumarappan Kathiresan, and S. Arockiaraj. "p-Wiener intervals and p-Wiener free intervals." Discussiones Mathematicae Graph Theory 32.1 (2012): 121-127. <http://eudml.org/doc/270806>.

@article{KumarappanKathiresan2012,
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.},
author = {Kumarappan Kathiresan, S. Arockiaraj},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {Wiener index of a graph; Wiener graphical; p-Wiener interval; p-Wiener free interval; hyper-Wiener index of a graph; radius; diameter; -Wiener interval; -Wiener free interval},
language = {eng},
number = {1},
pages = {121-127},
title = {p-Wiener intervals and p-Wiener free intervals},
url = {http://eudml.org/doc/270806},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Kumarappan Kathiresan
AU - S. Arockiaraj
TI - p-Wiener intervals and p-Wiener free intervals
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 1
SP - 121
EP - 127
AB - 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.
LA - eng
KW - Wiener index of a graph; Wiener graphical; p-Wiener interval; p-Wiener free interval; hyper-Wiener index of a graph; radius; diameter; -Wiener interval; -Wiener free interval
UR - http://eudml.org/doc/270806
ER -

References

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  1. [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley Reading, 1990). Zbl0688.05017
  2. [2] P.G. Doyle and J.L. Snell, Random Walks and Electric Networks (Math. Assoc., Washington, 1984). Zbl0583.60065
  3. [3] R.C. Entringer, D.E. Jackson and D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (101) 1976. Zbl0329.05112
  4. [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. Zbl0963.92015
  5. [5] I. Gutman, Y.N. Yeh, S.L. Lee and J.C. Chen, Wiener number of dendrimers, Comm. Math. Chem., 30 (1994) 103-115. Zbl0798.05052
  6. [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. [7] KM. Kathiresan and S. Arockiaraj, Wiener indices of generalized complementary prisms, Bull. Inst. Combin. Appl. 59 (2010) 31-45. Zbl1221.05116
  8. [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. Zbl1034.92040
  9. [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. Zbl1199.05091
  10. [10] S. Nikolić, N. Trinajstić and Z. Mihalić, The Wiener index: Development and applications, Croat. Chem. Acta. 68 (1995) 105-129. 
  11. [11] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69 (1947) 17-20, doi: 10.1021/ja01193a005. 

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