Discussiones Mathematicae Graph Theory 32(1) (2012) 121-127
doi: 10.7151/dmgt.1590

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p-Wiener Intervals and p-Wiener Free Intervals

Kumarappan Kathiresan

Center for Research and Post Graduate Studies in Mathematics
Ayya Nadar Janaki Ammal College
Sivakasi - 626 124,Tamil Nadu, INDIA

S. Arockiaraj

Department of Mathematics
Dr. Sivanthi Aditanar College of Engineering
Tiruchendur-628 215,Tamil Nadu, INDIA


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.


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Received 8 July 2010
Revised 15 February 2011
Accepted 15 February 2011