p-Wiener intervals and p-Wiener free intervals

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( 6= 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.