On the Wiener index of two families generated by joining a graph to a tree

The Wiener index W (G) of a graph G is the sum of distances between all vertices of G. The Wiener index of a family of connected graphs is defined as the sum of the Wiener indices of its members. Two families of graphs can be constructed by identifying a fixed vertex of an arbitrary graph F with vertices or subdivision vertices of an arbitrary tree T of order n. Let Gv be a graph obtained by identifying a fixed vertex of F with a vertex v of T . The first family V = {Gv | v ∈ V (T )} contains n graphs. Denote by Gve a graph obtained by identifying the same fixed vertex of F with the subdivision vertex ve of an edge e in T . The second family E = {Gve | e ∈ E(T )} contains n − 1 graphs. It is proved that W (V) = W (E) if and only if W (F ) = 2W (T ).


Introduction
In this paper we are concerned with undirected connected graphs G with vertex set V (G) and edge set E(G). The order n G of a graph G is the number of its vertices. By distance d G (u, v) between vertices u and v of G we mean the number of edges in a shortest path connecting them. Denote by d G (v) the sum of distances from a vertex v to all vertices of G. The Wiener index is a topological index based on distances between vertices of a graph G: It was introduced to quantify the structural information of tree-like molecular graphs [25] and have found numerous applications in organic chemistry and other fields. In particular, the Wiener index was used for the design of quantitative structure-property relations for chemical compounds [1,2,17,20,21]. The success in the application of this index has been the reason for the intensive study of its mathematical properties. The comprehensive bibliography regarding this index can be found in books [3,14,[22][23][24] and reviews [2, 9, 11-13, 15, 16, 18].
In this paper, we study the Wiener index for two families of graphs. These families arise as a result of structural transformations of a graph. The Wiener index of a family G of connected graphs G = {G 1 , G 2 , . . . , G k } is defined as the sum of the Wiener indices of its members, The subdivision of an edge e in a tree T is the replacement of e with a new path of order 3. The resulting tree T e contains one more edge than T . The central vertex v e of the path is called the subdivision vertex. Denote by P n , C n , S n , and S a,b,c the path, the cycle, the star of order n and the generalized star with branches of length a, b, c and order n = a + b + c + 1.
Two families of graphs can be constructed by identifying a fixed vertex of an arbitrary graph F with vertices or subdivision vertices in an arbitrary tree T . Let G v be a graph obtained by identifying a fixed vertex u of F with a vertex v of T . The first family V = {G v | v ∈ V (T )} contains n graphs of order n T + n F − 1 (see the left part of Figure 1). Denote by G ve a graph obtained by identifying the same fixed vertex u of F with the subdivision vertex v e of an edge e in T . The second family E = {G ve | e ∈ E(T )} contains n − 1 graphs with n T + n F vertices (see the right part of Figure 1). We are interesting in finding conditions for the coincidence the Wiener indices of families V and E. * E-mail address: dobr@math.nsc.ru

Main result
We need two useful results concerning Wiener index of the family of trees obtained from a tree by subdivisions of its edges. Subdivisions of edges of a tree T form a family containing trees T e , e ∈ E(T ). The following result was proved in [7]: T )} are obtained from an arbitrary tree T and a graph F as described above (see Figure 1). Proof. Let graph G v be obtained by identifying a vertex v ∈ V (T ) and a vertex u ∈ V (F ). Then W (G v ) can be presented as follows [19] W Summing this equation for all vertices v ∈ V (T ), we have Let graph G e be obtained by identifying the subdivision vertex v e of T e and a vertex u ∈ V (F ). Then Summing this equation for all edges v ∈ E(T ), we get Applying Lemmas 2.1 and 2.2, we can write As an illustration, consider two families of unicyclic graphs obtained from a tree T and a graph The above considerations may be useful for estimating how the Wiener index changes on average under attaching a graph F to a tree T . The resulting expressions do not depend on local distance characteristics of tree vertices: For the smallest graph F ∼ = P 2 , Values W avr (V) or W avr (E) may be integer. Coefficients at W (T ) are equal to 2 if n F = n T /2 + 1 or n F = n T /2 − 1 for even n T . Since W (P n ) = n(n − 1)(n + 1)/6, both quantities W avr (V) and W avr (E) are integers if T ∼ = P 6m−1 , m ≥ 1. Average values of the Wiener index may be realized by graphs. Let families V and E be constructed using T ∼ = P 7 , F ∼ = P 2 and T ∼ = P 9 , F ∼ = C 6 , respectively. Then W avr (V) = 79 and W avr (E) = 339 are realized by graphs of V and E.

Conclusion
In the presented paper, we have studied properties of the Wiener index for two families of graphs. Properties of this kind can be regarded as collective properties of the Wiener index, i.e. the main results don't reflect the property of the index of any particular graph, but a collective property of families of such graphs. This approach may be useful in studying of topological indices of mixtures of molecular graphs or in characterizing sets of graphs obtained after destruction of complex formations. Some results in this direction have been reported for trees and hexagonal chains in [4][5][6][7][8]10].