## Wiener index of trees theory and applications

GRAPH THEORY and APPLICATIONS Trees as Models. ▫ Trees are used in many applications: This sum is called the Wiener Index of G. ,. ( ). ( , ). G uv V. It was shown that the Wiener index of a tree and its line graph are always distinct I. Gutman, Wiener index for trees: theory and applications, Acta Appl. Math. graphs is of interest both in theoretical investigations and in applications. The stimulated the elaboration of the theory of Wiener indices of trees (for details,. 11 Jan 2016 The extremal Wiener polarity index of (chemical) trees with given different R. & Gutman, I. Wiener index of trees: theory and applications. The field of graph theory is rich in its theoretical and application area. R. Entringer and I. Gutman, Wiener index of trees: theory and applications, Acta Appl. 20 Oct 2014 Wiener index of trees: Theory and applications. Acta. Appl. Math. 2001, 66, 211– 249. 34. Hamzeh, A.; Iranmanesh, A.; Reti Graph Theory 36 (2016) 455–465 the case of trees. An application in chemistry of the Steiner Wiener index is one of the basic concepts of graph theory [4].

## 22 Jun 2017 trees. Keywords: Distance in graphs, Wiener index, peripheral Wiener [1] J.A. Bondy and U.S.R. Murty, Graph theory with applications, The

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 The Wiener index is defined as the sum of all distances between vertices of the graph under consideration. For more information on the Wiener index, the and second-minimum Wiener indices among all the trees with n vertices and diameter d The vast majority of chemical applications of the Wiener index deal with [4] A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and. 12 Jan 2016 Already in 1947, Wiener has shown that the Wiener index of a tree can be and I . Gutman, Wiener index of trees: Theory and applications,.

### 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

The Wiener index is also closely related to the closeness cen- trality of a vertex in and has been frequently used in sociometry and the theory of social networks [ 2]. and chemical applications, International Journal of ChemTech. Research 22 Jun 2017 trees. Keywords: Distance in graphs, Wiener index, peripheral Wiener [1] J.A. Bondy and U.S.R. Murty, Graph theory with applications, The Applications of the hyper-Wiener index as well as its calculation are [1] A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications 2 Nov 2010 The Wiener index of a graph G, denoted by W(G), is the sum of distances between all pairs of The concept of line graph has various applications in physical chemistry [12, 15]. Recently Wiener index of trees: theory and.

### Graph Theory 36 (2016) 455–465 the case of trees. An application in chemistry of the Steiner Wiener index is one of the basic concepts of graph theory [4].

Graph Theory 36 (2016) 455–465 the case of trees. An application in chemistry of the Steiner Wiener index is one of the basic concepts of graph theory [4]. The Wiener index is also closely related to the closeness cen- trality of a vertex in and has been frequently used in sociometry and the theory of social networks [ 2]. and chemical applications, International Journal of ChemTech. Research 22 Jun 2017 trees. Keywords: Distance in graphs, Wiener index, peripheral Wiener [1] J.A. Bondy and U.S.R. Murty, Graph theory with applications, The Applications of the hyper-Wiener index as well as its calculation are [1] A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications 2 Nov 2010 The Wiener index of a graph G, denoted by W(G), is the sum of distances between all pairs of The concept of line graph has various applications in physical chemistry [12, 15]. Recently Wiener index of trees: theory and. For trees and unicyclic graphs, distance energy is equal to the doubled value of The huge majority of chemical applications of the Wiener index deal with acyclic From the theory of linear recurrence equations, there exist constants ak and applications. To give one have the greatest Wiener index among trees with n vertices and m segments. 2 Wiener index of trees: Theory and applications.

## The Wiener index is a graph invariant defined as the sum of distances between all vertices of \(G\): $$ W(G)= \sum_{u,v \in V(G)} d(u,v)=\frac12 \sum_{v \in V(G)} d_G(v). $$ It was introduced as a structural descriptor for tree-like molecular graphs [ 1 ].

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 Wiener Index of Trees: Theory and Applications. Andrey A. Dobrynin ,; Roger Entringer &; Ivan Gutman. 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 The Wiener index is defined as the sum of all distances between vertices of the graph under consideration. For more information on the Wiener index, the and second-minimum Wiener indices among all the trees with n vertices and diameter d The vast majority of chemical applications of the Wiener index deal with [4] A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: theory and. 12 Jan 2016 Already in 1947, Wiener has shown that the Wiener index of a tree can be and I . Gutman, Wiener index of trees: Theory and applications,.

Connectivity parameters have important role in the study of networks in the physical world. Wiener index is one such parameter with several applications in chemistry and network theory. In this article Wiener index of various fuzzy graph structures like fuzzy trees and fuzzy cycles are discussed. Relationship between connectivity index and Wiener index of a fuzzy graph is also studied. The Wiener index is a graph invariant defined as the sum of distances between all vertices of \(G\): $$ W(G)= \sum_{u,v \in V(G)} d(u,v)=\frac12 \sum_{v \in V(G)} d_G(v). $$ It was introduced as a structural descriptor for tree-like molecular graphs [ 1 ]. 2. Variation of the Wiener index under tree transformations. Let T be a tree with vertices u, v ∈ V (T) such that k = d (u, v) and π = u 0 u 1 ⋯ u k the path in T which connects u = u 0 and v = u k. Then for each 0 ⩽ i ⩽ k we define the sets N u i (π) = {x ∈ V (T): u i ∈ π (x, u) ∩ π (x, v)}, where π (x, u) denotes the path connecting x and u. The essential part of this paper deals with trees homeomorphic to the graph H on 6 vertices, depicted in Fig. 1, that is, the trees that have precisely two vertices of degree 3, four vertices of degree 1 and all other vertices of degree 2.(Recall that graphs G 1 and G 2 are homeomorphic if and only if the graphs obtained from G 1 and G 2, respectively, by repeatedly substituting the vertices In chemistry, graph invariants are known as topological indices. Topological indices have many applications as tools for modeling chemical and other properties of molecules. The Wiener index is one of the most studied topological indices, both from a theoretical point of view and applications.