The WIENER INDEX W and the edge-WIENER INDEX W_e of G are defined as the sum of distances between all pairs of vertices in G and the sum of distances between all pairs of edges in G, respectively. In this paper, we identify the four trees, with the first through fourth greatest WIENER and edge-WIENER INDEX among all trees of order n ≥ 10.

Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and bonds by the edges of this graph. Topological graph indices have been successfully used in determining the structural properties and in predicting certain physicochemical properties of chemical compounds. WIENER INDEX is the oldest topological INDEX which can be used for analyzing intrinsic properties of a molecular structure in chemistry. The WIENER INDEX of a graph G is equal to the sum of distances between all pairs of vertices of G. Recently, the entire versions of several indices have been introduced and studied due to their applications. Here we introduce the entire WIENER INDEX of a graph. Exact values of this INDEX for trees and some graph families are obtained, some properties and bounds for the entire WIENER INDEX are established. Exact values of this new INDEX for subdivision and k-subdivision graphs and some graph operations are obtained.

Please click on PDF to view the abstract.

The WIENER INDEX W (G) of a connected graph G is defined as W (G) = Su, v Î v (G) dG (u, v) where dG (u, v) is the distance between the vertices u and v of G. For S Í V (G), the Steiner distance d (S) of the vertices of S is the minimum size of a connected subgraph of G whose vertex set is S. The k-th Steiner WIENER INDEX SWk (G) of G is defined as SWk (G) = S SÍV (G)÷s÷=k d(S). We establish expressions for the k-th Steiner WIENER INDEX on the join, corona, cluster, lexicographical product, and Cartesian product of graphs.

Let G be an (n, m)-graph. We say that G has property (*) if for every pair of its adjacent vertices x and y, there exists a vertex z, such that z is not adjacent to either x or y. If the graph G has property (*), then its complement G- is connected, has diameter 2, and its WIENER INDEX is equal to (n2) + m, i.e., the WIENER INDEX is insensitive of any other structural details of the graph G. We characterize numerous classes of graphs possessing property (*), among which are trees, regular, and unicyclic graphs.

Let R be a commutative ring and G (R) be its zero-divisor graph. In this article, we study WIENER INDEX and energy of G (Zn) where n=pq or n=p2q and p, q are primes. A MATLAB code for our calculations is also presented.

Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. For a subset $S$ of $V(G)$, the Steiner distance $d(S)$ of $S$ is the minimum size of a connected subgraph whose vertex set contains $S$. For an integer $k$ with $2 \le k \le n - 1$, the $k$-th Steiner WIENER INDEX of a graph $G$ is defined as $SW_k(G) = \sum_{\substack{S\subseteq V(G)\\ |S|=k}}d(S)$. In this paper, we present exact values of the $k$-th Steiner WIENER INDEX of complete $m$-ary trees by using inclusion-excluision principle for various values of $k$.