Geodesic
p-Wiener intervals and p-Wiener free intervals
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 121-127

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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.
Keywords: Wiener index of a graph, Wiener graphical, p-Wiener interval, p-Wiener free interval, hyper-Wiener index of a graph, radius, diameter 
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Kathiresan, Kumarappan; Arockiaraj, S. p-Wiener intervals and p-Wiener free intervals. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 121-127. http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a9/