Detour Hosoya Polynomials of Some Compound Graphs

In this paper we will introduce a new graph distance based polynomial; Detour Hosoya polynomials of graphs ) x ; G ( H * . The Detour Hosoya polynomials ) x ; G ( H for some special graphs such as paths and cycles are obtained. Moreover the Detour Hosoya polynomials ) x ; G G ( H 2 1 • , 1 2 * H ( G : G ; x ) and 1 2  * H ( G G ; x ) are obtained.

In this paper, we consider finite connected graphs without loops or multiple edges.For undefined concepts and notations see [9] and [12].
Ordinarily, when we wish to proceed from a point A to a point B we take a route which involves the least distance.We have all been faced with detour sign which require us to take a route from A to B that involves a greater distance.In any such detour route from A to B we assume that there is no possible shortcut along the route, for otherwise this should have been part of the route initially.When one is driving along such a detour, it sometimes seems that we are using the longest route possible from A to B (again subject to the "no shortcut" condition).In this paper we investigate longest detour routes in graphs.
is the number of pairs of vertices in G with detour distance k, and   In 1993, Gutman [8], established few additional properties of the respective graph polynomials.He obtained Hosoya polynomials of some special graphs and obtained formula for the Hosoya polynomials of some compound graphs, namely G will be obtained.

Detour Hosoya Polynomials of Some Special Graphs
Let n P , n K and n S denotes the path, complete and star graphs of n vertices respectively.It is known that [10] all trees and complete graphs are detour graphs.This leads us to the following result.We know that [11], for an odd p, the ordinary Hosoya polynomial of .
The following result gives us the Wiener index of the detour distance of the special graphs n  It is clear that  .We will consider the following cases:

[ 10 ]
two vertices u and v in a connected graph G is the length of a shortest u-v path in G.For a nonempty set S of vertices of G, the subgraph   S of G induced by S as its vertex set while an edge of G belongs to   S if it joins two vertices of S. If P is a u-v vertices of P is P itself.This observation suggests the following concept.u and v in G is the length of a longest induced u-v path, that is a longest u-v path P for which if u and v are adjacent.Also, note that u and v of G. Therefore the detour distance is symmetric.However, the triangle inequality does not hold in general.Consider the wheel general, the detour distance is not a metric on the vertex set of Ggraph G is the set consisting of all detour eccentricities of G, that is u and v of G.No cycle of length 5 or more is a detour graph.On the other hand, all trees and all complete graphs are detour graphs.If u and v are distinct vertices of a graph G such that , the converse is not true in general, that is if the number of pairs of vertices in G that are distance k apart, and ) G (  is the diameter of the graph G .In this paper, the concept of Hosoya polynomials of detour distance of a connected graph G ( or simply detour Hosoya polynomial of a graph G) of detour distances between all pairs of vertices of the graph G is known as the Wiener index of detour distance of the graph G (or simply detour Wiener index of the graph G), that is ideas in the following example.Example 1.1.Let G be a graph of order 9 p = ,depicted in figure 1.1(a).

1 G and 2 G 1 G and 2 GG and 2 G
be vertex-disjoint connected graphs, and let ) by identifying the two vertices u and v.This means that 1 have exactly one vertex in common in the compound graph joining the two vertices u and v.In this paper, formulas for

Proposition 2
Let u,v be any two distinct vertices of p C .We will consider the following cases:

Proposition 2 . 3
prove the formula for the case when p is even.This completes the proof.■ Let p W be a wheel graph of 6

. 3 .
Detour Hosoya Polynomials of Some Compound GraphsLet u be a vertex of a connected graph G of order p .The number of pairs of vertices of G containing the vertex u such that k

Theorem 3 . 2 1 G and 2 G
P will contain the vertex w.If P  is longest ) of P  , other than w is adjacent with a vertex of P in the Cases (2) and (3).Now, adding the polynomials obtained from the cases (1), (2) and (3), we get the required result.■ If are disjoint connected graphs, then