On Graphs with Same Distance Distribution

In the present paper we investigate the relationship between Wiener number W , hyper-Wiener number R , Wiener vectors WV , hyper-Wiener vectors HWV , Wiener polynomial H , hyper-Wiener polynomial HH and distance distribution DD of a (molecular) graph. It is shown that for connected graphs G and * G , the following five statements are equivalent: 1) ( ) ( ) * DD G DD G = , 2) ( ) ( ) * WV G WV G = , 3) ( ) ( ) * HWV G HWV G = , 4) ( ) ( ) * H G H G = , 5) ( ) ( ) * HH G HH G = ; and if G and * G have same distance distribution DD then they have same W and R but the contrary is not true. Therefore, we further investigate the graphs with same distance distribution. Some construction methods for finding graphs with same distance distribution are given.


Introduction
The Wiener index is one of the oldest topological indices of molecular structures.
It was put forward by the physico-chemist Harold Wiener [1] in 1947.The Wiener index of a connected graph G is defined as the sum of distances between all pairs of vertices in G : As an extension of the Wiener index of a tree, Randić [2] introduced Wiener matrix W and hyper-Wiener index R of a tree.For any two vertices , i j in . In Refs.[3] [4], Randic and Guo and colleagues further introduced the higher Wiener numbers and some other Wiener matrix invariants of a tree T .The higher Wiener numbers can be represented by a Wiener number sequence After the hyper-Wiener index of a tree was introduced, many publications [5]- [11] have appeared on calculation and generalization of the hyper-Wiener index.Klein et al. [5] generalized the hyper-Wiener index so as to be applicable to any connected structure.Their formula for the hyper-Wiener index R of a graph G is The relation between Hyper-Wiener and Wiener index was given by Gutman [11].
The Hosoya polynomial

( )
H G of a connected graph G was introduced by Hosoya [12] in 1988, which he named as the Wiener polynomial of a graph: where ( ) , d G k is the number of pairs of vertices in the graph G that are distance k apart.
In Ref. [13], Cash introduced a new hyper-Hosoya polynomial The relationship between the Hosoya polynomial and the Hyper-Hosoya polynomial was discussed [13].
The sequence  is also known (since 1981) as the dis- tance distribution of a graph G [14], denoted by ( ) ∑ .Later the definition of higher Wiener numbers is extended to be applicable to any connected structure by Guo et al. [15].For a connected graph G with n vertices, denoted by 1, 2, , n where ij d is the distance between vertices i and j .Then called the higher Wiener numbers of G .The vector ( ) , , W W  is called the hyper-Wiener vector of G , denoted by ( ) HWV G .The concept of the Wiener vector of a graph is also introduced in ref. [15].For a connected graph G with n vertices, denoted by 1, 2, , n  , let , , W W  is called the Wiener vector of G , denoted by ( ) WV G .Moreover, a matrix sequence , , , W W W  , called the Wiener matrix sequence, and their sum , called the hyper-Wiener matrix, are introduced, where is the distance matrix.A Wiener polynomial sequence and a weighted hyper Wiener polynomial of a graph are also introduced.
In this paper, based on the results in ref. [15], we study the relation between Wiener number W , hyper-Wiener number R , Wiener vector WV , hyper-Wiener vector HWV , Hosoya polynomial H , hyper-Hosoya polynomial HH and distance distribution DD of a graph.It is shown that for connected graphs G and * G , the the contrary is not true.This means that the distance dis- tribution of a graph is an important topological index of molecular graphs.Therefore, we further investigate the graphs with same distance distribution.It is shown that the graphs with same vertex number, edge number, and diameter 2 have same distance distribution.Some construction methods for finding graphs with same distance distribution are given.
(1)⇒ (2).By the definitions of DD and WV , ( Assume, for ( )  , and so )⇒ (5).By the definitions of Hosoya polynomial H and hyper-Hosoya polynomial HH , it is easy to see that, if ( However, the contrary of the theorem doesn't hold.For instance, the trees 1 T

Graphs with Same Distance Distribution
From the above theorems, one can see that, if two graphs G and * G have  same distance distribution DD , then they have same , , , , W WW WV HWV H and HH .So it is significant to study the graphs with same distance dis-tribution.
1) The minimum non-isomorphic acyclic graphs with same DD By direct calculation, we know the minimum non-isomorphic acyclic graphs with same distance distribution are the following two pairs of trees in Figure 2 which have 9 vertices.
2) The minimum non-isomorphic cyclic graphs with same DD The minimum non-isomorphic cyclic graphs with same distance distribution are the following graphs with 4 vertices (see Figure 3).Note that, for the above graphs with same distance distribution, their Wiener    ( ) ( ) V denote the graph obtained from vertex-disjoint graphs G and H by connecting every vertex of G to every vertex of H .
Corollary 3.3.Let G G V have same distance distribution.
. For graphs with diameter greater than or equal to 2, we will give some construction methods for finding graphs with same distance distribution.
Let G be a connected graph with vertices set { } , and let be the distant matrix of the graph G. Let ( ) , , , G′ ) be the connected graphs with 1 n (resp. 2 n ) vertices and with same distance distribution.For ( ) ( ) , let G (resp.* G ) be the graph ob-tained from 1 G and 1 G′ (resp. 2 G and 2 G′ ) by identifying 1 v and 1 v′ (resp. 2 v and 2 v′ ).If ( ) ( ) ( ) ( ) , then G and * G have same distance distribution.

DD G DD G =
. Let H be a connected graph vertex-disjoint with 1 G and 2 G .For ( ) , then G and * G have same distance distribution.From Theorem 3.5, one can obtain graphs with same distance distribution in Figure 4 shows two pairs of graphs with 5 vertices and 5 edges and with same DD , one of which has diameter 2 and the other has diameter 3.
Figure 5 shows three pairs of graphs with 6 vertices and 6 edges and with   It is easy to see that the graphs in Figure 5 can be obtained from graphs in Figure 3, Figure 4 by the construction methods given in Theorems 3.4, 3.5.
However, the construction methods are not complete.There might be some graphs with same DD which could not be obtained by the above construction methods.
Open Problem.Is there a construction method for finding all graphs with same distance distribution?
V G is the vertex set of G , and ( ) , G d u v is the distance between vertices u and v in G .
G denote the diameter of a graph G .Theorem 2.1.Let G and * G be connected graphs.Then the following five statements are equivalent: 1) G and * G have same distance distribution DD ; 2) G and * G have same Wiener vector WV ; 3) G and * G have same hyper-Wiener vector HWV ; 4) G and * G have same Wiener polynomial H ; 5) G and * G have same hyper-Wiener polynomial HH .
2.2.Let G and * G be two graphs with same distance dis- tribution.Then G and * G have same W and R .Proof: By the definitions of DD , W and R , (

and * 1 T 2 T
(resp. 2 T and * ) in Figure 1 have same W and R , but they have different distance distributions.
matrix sequences and hyper-Wiener matrices are different.The following theorem gives a class of graphs with same distance distribution.Let , n m  be the set of all the graphs with n vertices and m edges.G = = , we have


with same distance distribution by adding a new vertex and some edges.