let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...

The Wiener number of a graph G is defined as 1 2 ∑ u,v∈V (G) d(u, v), d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, W (μ(S n)) ≤ W (μ(T k n )) ≤ W (μ(P k n )), where Sn, Tn and Pn denote a star, a gen...

The Wiener number of a graph G is defined as 1 2 ∑ d(u, v), where u, v ∈ V (G), and d is the distance function on G. The Wiener number has important applications in chemistry. We determine the Wiener number of an important family of graphs, namely, the Kneser graphs.

Let T be an acyclic molecule with n vertices, and let S(T ) be the acyclic molecule obtained from T by replacing each edge of T by a path of length two. In this work, we show that the Wiener index of T can be explained as the number of matchings with n− 2 edges in S(T ). Furthermore, some related results are also obtained.

We present explicit formulae for the eccentric connectivity index and Wiener index of 2-dimensional square and comb lattices with open ends. The formulae for these indices of 2-dimensional square lattices with ends closed at themselves are also derived. The index for closed ends case divided by the same index for open ends case in the limit N →&infin defines a novel quantity we call compression...

An explicit, non-recursive formula for the Wiener index of any given benzenoid chain is derived, greatly speeding up calculations and rendering it manually manageable, through a novel envisioning of chains as ternary strings. Previous results are encompassed and two completely new and useful ones are obtained, a formula to determine Wiener indices of benzenoid chains in periodic patterns, and a...

Hansen et. al., using the AutoGraphiX software package, conjectured that the Szeged index Sz(G) and the Wiener index W (G) of a connected bipartite graph G with n ≥ 4 vertices and m ≥ n edges, obeys the relation Sz(G) − W (G) ≥ 4n − 8. Moreover, this bound would be the best possible. This paper offers a proof to this conjecture.

We study distance-based graph invariants, such as the Wiener index, the Szeged index, and variants of these two. Relations between the various indices for trees are provided as well as formulas for line graphs and product graphs. This allows us, for instance, to establish formulas for the edge Wiener index of Hamming graphs, C4nanotubes and C4-nanotori. We also determine minimum and maximum of ...

Molecules and molecular compounds are often modeled by molecular graphs. One of the most widely known topological descriptor [6, 9] is the Wiener index named after chemist Harold Wiener [13]. The Wiener index of a graph G(V, E) is defined as W (G) = ∑ u,v∈V d(u, v), where d(u, v) is the distance between vertices u and v (minimum number of edges between u and v). A majority of the chemical appli...