Hosoya Polynomials of Generalized Vertex Identified and Edge Introducing Graphs

تانايبلل تافاحلا ثادحتساو سوؤرلا قباطت ميمعتل ايوسوه دودح تاددعتم  يلع دمحم دمحأ  شاهد دولوم رون  ةيلك  ع تايضايرلاو بوساحلا مول  لصوملا ةعماج قا رعلا ،لصوملا ،  :ثحبلا ملاتسا خيرات 13 / 05 / 2012  :ثحبلا لوبق خيرات 18 / 09 / 2012  صخلملا  لل ل سو رللل ة لبنلاا نيللصفنمو نيلصتم نينايب يلأ تانايبلل ةفاح ةفاضإو سأ رلا قباطت ةيلمع ميمعت مت n  ( 3  n .سو رلل ة بنلاا اهنيب اميف ةلصفنملاو ةلصتملا تانايبلا نم )  ةل رملا تالنايبلا اذله للبمل ةفالبملا ةتلعمو رلنيو لليل و ايولسوه وتلح تا تلعتم مللع انللصح ثح لا اذه يف يتغيص ملع امتعلااا Gutman's  ملع يضايرلا ءا رقتسلاا ماتختسابو n .  :ةيحاتفملا تاملكلا .رنيو ليل ،ايوسوه وتح ة تعتم ،ةفابملا


1.Introduction
Our terminology and notations will be as given in the references [2,3,5]. Let G be a connected graph of order p and size q . The distance between the two vertices u and v in G is the minimum of the lengths of v u − paths in G ;it is denoted by is the vertex set of G . The diameter of G denoted by G diam , or , is defined by: The concept of distance based polynomial was introduced in 1988 by H. Hosoya [7]. Hosoya's paper, which seems to be the only published work concerning this issue, reported a limited number of results on Hosoya polynomials. Let ) , ( k G d be the number of pairs of vertices in G that are at distance k apart , , then the Hosoya polynomial of a connected graph G is defined as follows : The number of all pairs of vertices u and v in G is 2 be the number of vertices in G , that are at distance k from vertex v , then the Hosoya polynomial of a vertex v is defined by [4]   = 0 ) , , Observe that From this ,we have a relation between the Hosoya polynomial of G and Hosoya polynomial of a vertex v in G , namely, Several authers had obtained Hosoya polynomials for special graphs and compound graphs obtained by certain graph binary operations [7,8,10,11]. In 1993, Gutman [6] defined vertex identified and edge introducing graphs, constructed from two vertex disjoint connected graphs as given next: . ■ In this paper, we continue the research along the same lines to find Hosoya polynomials and Wiener indices of graphs constructed from several vertex disjoint graphs by using the idea of vertex identification and edge introducing.

Generalized Vertex Identified Graphs:
In the next theorem, we give the Hosoya polynomial of n J for all 2  n .
. We assume that (2.1) is true for 2  = r n and prove it for 1 + = r n . Then, for the graph using Gutman's Theorem, we get ) ; ( ) ; ( Simplifying the last expression, we obtain (2.3). ■ If each of

Corollary 2.4: If G is a connected graph and
, ( , then for any positive integer n ,

Corollary 2.5:
If G is a connected graph and

Generalized Edge Introducing Graphs
Let n G G G , ... , , 2 1 be disjoint connected graphs and let ) ( ,  It is clear that , and , and prove that it is true for n=r+1. ) ; ,  . ■ Now, using Theorem 3.1 and Lemma 3.2 , we prove the following theorem .
Proof: From Theorem 3.1, we have Then, from Lemma 3.2,we obtain is denoted by

Corollary 3.4: If G is a connected graph and
, then for any positive integer n, 3  n ,we obtain.

Let t
S be the star of order t , then assuming that the vertex of identification of each t S is an end vertex and by using Corollary 2.5, we get 3. Let t W be the wheel graph of order t, Then , assuming that the vertex of identification of each t W is vertex of degree 3 and by using Corollary 2.5 , we get.

Hosoya Polynomials of Special Edge Introducing Graphs
To illustrate the usefulness of Corollary 3.5,we take G to be a special graph ,such as complete graph, even cycle, star or wheel.
 is a star of order t, then assuming that the vertex of introducing edge of each t S is an end vertex and by using Corollary 3.5,we get 3. If t W , 4 t  , is a wheel of order t then , assuming that the vertex of introducing edge of each t W is a vertex of degree 3 and by using Corollary 3.5, we get

Wiener Index and Average Distance.
The Wiener index of a connected graph G is denoted by W(G) and defined by The name Wiener index for the quantity defined in (4.3.1) is usual in chemical literature, since Harold Wiener [8] in 1947 seems to be the first who considered it. Several mathematical authors obtained the Wiener index of many kinds of chain of cycle graphs [2,10] .The Wiener index of G can be obtained from the following formula Finally , we define the Wiener index of any vertex u in a connected graph G by , Now, from Corollary (4.3.1), we obtain the following corollary.

5.Conclusions
In this paper, we generalized the operation of constructing vertex identified and edge introducing graphs obtained from two graphs to those obtained from a sequence of n pairwise vertex disjoint connected graphs n G G G ,...., , 2 1 . Using Gutman's Theorems n times in each case, we obtained Hosoya polynomials of such generalized graphs.
Moreover, we obtained Hosoya polynomials, Wiener indices and average distance for generalized vertex identified and edge introducing graphs when every , 1 , n i G i   is isomorphic to a special graph which has important applications in Chemistry.