## Wiener index in graph theory

The Wiener index of a graph G = (V, E), denoted by W (G), was introduced in 1947 by chemist Harold Wiener as the sum of distances between all vertices of G: W (G) = ∑ {u, v} ⊆ V (G) d (u, v). The first and the second Zagreb indices were introduced more than thirty years ago by Gutman and Trinajestic.

The Wiener index of a graph G = (V, E), denoted by W (G), was introduced in 1947 by chemist Harold Wiener as the sum of distances between all vertices of G: W (G) = ∑ {u, v} ⊆ V (G) d (u, v). The first and the second Zagreb indices were introduced more than thirty years ago by Gutman and Trinajestic. “The Wiener index of a graph is represented by and defined as the sum of distances between all pairs of vertices in a simple graph ”: Based on the Wiener index, Hosoya introduced the Wiener polynomial (now called Hosoya polynomial) in 1988 [ 8 ]. Abstract. The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and Wiener index of trees As the path and therefore the distance between two vertices of a tree is unique, the Wiener index of a tree is much easier to compute than that of an arbitrary graph. The Wiener index or Wiener number W(G) of G is deﬁned as W (G)= 1 2 u∈V (G) v∈V (G) d G (u,v). (1.1) Here, d G (u,v) (or simply d(u,v) when no confusion arises) denotes the distance between the vertices u and v in G (for standard deﬁnitions and notations in graph theory see [BR00], [GY06], [Har69] or [Wes96]). Given a vertex u ∈ V(G), we deﬁne d+(u,G)as In chemical graph theory, the Wiener index (also Wiener number) is a topological index of a molecule, defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule. The Wiener index of a graph G is defined to be 2 , ( ) ( , ), u V G d u ∈ ∑ X X where d(u, X) is the distance between the vertices u and X in G. In this paper, we obtain an explicit expression for the Wiener index of an odd graph.

## Let G be a connected graph whose vertex and edge sets are V(G) and tween Wiener index and graph connectivity have not been established so far. Our aim is [1] J. А. В о n d y, U. S. R. M u г t y, Graph Theory with Applications , Macmilla.

Terminal Wiener Index for Graph Structures. [4] A.A. Dobrynin, R. Entringer, I. Gutman, “Wiener Index for trees: theory and applications”, Acta Appl. Math.,  05C12, 92E10. Keywords: Graph, Molecules, distance matrix, Wiener Index, Platt number lines are named edges in graph theory language. The advantage   It is well-known that an important domain in chemical graph theory is distance for computing the Wiener index of a graph have been proposed in the chemical. ing vertex of degree 4 and whose Wiener index equals the Wiener index [2] A.A. Dobrynin, Distance of iterated line graphs, Graph Theory Notes of New. The oldest topological index is the Wiener index, it has been extensively studied in many applications such as chemical graph theory, complex network, social

### Wiener Index The Wiener index, denoted (Wiener 1947) and also known as the "path number" or Wiener number (Plavšić et al. 1993), is a graph index defined for a graph on nodes by where is the graph distance matrix.

1 Jan 1995 Wiener Index Extension by Counting Even/Odd Graph Distances. of Structural Descriptors Derived from Information-Theory Operators. 3 Oct 2006 The Wiener index of a graph G is defined to be 1. 2 obtain an explicit expression for the Wiener index of an odd graph. A. A. Dobrynin, R. Entringer, and I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. a connected graph is known as the Wiener index of the graph. In this graphs on r vertices of Kn. Let G(n, r, k) be the graph obtained from complete I. Gutman, Y. Yeh, S. Lee, Y. Luo, Some recent results in the theory of Wiener number,. only mathematically computation of the edge Wiener index in certain graphs by this method. First we need some concepts from the theory of groups and graph

### It is well-known that an important domain in chemical graph theory is distance for computing the Wiener index of a graph have been proposed in the chemical.

The Mean Wiener index (denoted W) is the average of the distances between all pairs of vertices in a graph. For a graph having n vertices, We previously showed the Wiener number for 2,2,4-trimethylpentane is 66. Using this, we calculate that W = 66/28 = 2.35714. In graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge. The neighbourhood of a vertex v in a graph G is the subgraph of G induced by all vertices adjacent to v, i.e., the graph composed of the vertices adjacent to v and all edges connecting vertices adjacent to v. For example, in the image to

## In chemical graph theory, the Wiener index (also Wiener number) is a topological index of a molecule, defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule.

Terminal Wiener Index for Graph Structures. [4] A.A. Dobrynin, R. Entringer, I. Gutman, “Wiener Index for trees: theory and applications”, Acta Appl. Math.,  05C12, 92E10. Keywords: Graph, Molecules, distance matrix, Wiener Index, Platt number lines are named edges in graph theory language. The advantage   It is well-known that an important domain in chemical graph theory is distance for computing the Wiener index of a graph have been proposed in the chemical. ing vertex of degree 4 and whose Wiener index equals the Wiener index [2] A.A. Dobrynin, Distance of iterated line graphs, Graph Theory Notes of New. The oldest topological index is the Wiener index, it has been extensively studied in many applications such as chemical graph theory, complex network, social  5 May 2015 The hyper-Wiener index is one of distance-based graph invariants, Some recent results in the theory of the Wiener number, Ind. J. Chem 32  In chemical graph theory, the Wiener index (also Wiener number) introduced by Harry Wiener, is a topological index of a molecule, defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule.

graph. Some calculation results on the Wiener complexity and the Wiener index of (This article belongs to the Special Issue Graph Theory at Work in Carbon  The edge-Wiener index We (G) of a simple connected graph G is defined as the sum of distances between all pairs of edges of G. The main goal of this survey is   Terminal Wiener Index for Graph Structures. [4] A.A. Dobrynin, R. Entringer, I. Gutman, “Wiener Index for trees: theory and applications”, Acta Appl. Math.,  05C12, 92E10. Keywords: Graph, Molecules, distance matrix, Wiener Index, Platt number lines are named edges in graph theory language. The advantage   It is well-known that an important domain in chemical graph theory is distance for computing the Wiener index of a graph have been proposed in the chemical. ing vertex of degree 4 and whose Wiener index equals the Wiener index [2] A.A. Dobrynin, Distance of iterated line graphs, Graph Theory Notes of New.