Hosoya Polynomials of Steiner Distance of the Sequential Join of Graphs

The Hosoya polynomials of Steiner n-distance of the sequential join of graphs J3 and J4 are obtained and the Hosoya polynomials of Steiner 3distance of the sequential join of m graphs m J are also obtained.


Introduction
We follow the terminology of [2,3].For a connected graph of order p, the Steiner distance [5,6,7] of a non-empty subset , denoted by , is defined to be the size of the smallest connected subgraph T(S) of G that contains S; T(S) is a tree called a Steiner tree of S. If |S|=2, then d(S) is the distance between the two vertices of S. For … (1.4) Obviously, for Ali and Saeed [1] were first who studied this distance-based graph polynomial for Steiner n-distances, and established Hosoya polynomials of Steiner n-distance for some special graphs and graphs having some kind of regularity, and for Gutman's compound graphs , and defined by , as depicted in the following figure.
One can easily see that for In [8], Saeed obtained the (ordinary) Hosoya polynomials of m J , and in [7], Herish obtained the Steiner n-diameter of the sequential join of m empty graphs and of m complete graphs.Also, the Hosoya polynomials of Steiner distance of the sequential join of m empty graphs and of m complete graphs were obtained.For m 3  and n 2  , the Steiner n-diameter of the sequential join of m complete graphs is given by [7] where  is the smallest integer such that It is obvious that Eq. 1.6 holds for the sequential join of m graphs m J .
In this paper, a generalization of the results obtained in [7] is given.We obtained the Hosoya polynomials of Steiner n-distance of

Hosoya Polynomials of Steiner n-Distance of J 3 and J 4
In this section, we consider m J , for m=3 and m=4.Let S be any nsubset of vertices of m J .Let Proposition 2.1.For n p p p p , and ( ), ( ) and BG 3 () are as defined above.

Proof. It is clear that
.
in which C1 is the number of all n-subsets of V(J3) with Steiner distance equals n-1, and C2 is the number of all n-subsets of V(J3) with Steiner distance equals n.Therefore, This completes the proof.
The following corollary computes the n-Wiener index of J , and The number of these subsets S is given by . The number of these n-subsets is given by 1), ( 2), ( 3) and ( 4), we get C 1 as given in the statement of the proposition.

(II) d S n
This completes the proof.

Remark. The triple summation in C
1 is taken to be zero when n=3.

Hosoya Polynomials of
is the number of non-identical triangles K 3 as a subgraph in i G .
Proof.Let S be any 3-subset of vertices of m J , then we have three main cases for the subset S .= , when S is a connected subgraph in i G , and by Lemma 3.4.4. of [7], the number of such 3-subsets S is given by = , when S is a disconnected subgraph in i G , and the number of such 3-subsets S is given by

Case(I) produces the polynomial
.
(II) Either two vertices of S are in i V and one vertex of S in j V , i j  , or one vertex of S in i V , and two vertices of S in j V , for . For each such cases of S, , and the number of ways of choosing such S is given by

1 . 1 )
|S|=n, the Steiner distance of S is called Steiner n-distance of S in G.The Steiner n-diameter of G, denoted by It is clear that (If nm  , then sum of the Steiner n-distances of all n-subsets containing v. The sum of Steiner n-distances of all n-subsets of of n-subsets of distinct vertices of G with Steiner n-distance k.The graph polynomial defined by is the Steiner n-diameter of G; is called the Hosoya polynomial of Steiner n-distance of G.It is clear that of n-subsets S of distinct vertices of G containing u with Steiner n-distance k.It is clear that polynomials of G1 and G2.

Fig. 1
Fig. 1.1 m J J 3 and J 4 ; and the Hosoya polynomials of Steiner 3-distance of m J , number of all nsubsets S such that S is connected in i G .The following proposition determines the Hosoya polynomials of Steiner n-distance of J 3 .

Steiner 3 -J ( m 5 Theorem 3 . 1 .
Distance of m )In this section, we consider m following theorem determines Hosoya polynomials of Steiner 3-distance of m J .For m 5  , and (III), we get the required result.The numbers A and B are given in Theorem 3.1 can be counted when i i therefore the Hosoya polynomials of Steiner n-distance of J 4 has the following form iG .The number of these n-subsets is given by 

Example 2.6. Let p K 1 , p K 2 and p K 3 be complete graphs of orders p 1 , p
3 are given in Proposition 2.3.Remark.For m 5  , the calculation of the coefficients of n m H J x * ( ; 1 () , BG 2 () and BG 3 () are given in Proposition 2.1 can be counted for some specific graphs G 1  , P 2  and P 3  be path graphs of orders 1  ,