The n-Hosoya Polynomials of the Square of a Path and of a Cycle

The n-Hosoya polynomial of a connected graph G of order t is defined by:

is the number of pairs (v,S), in which . In this paper, we find the n-Hosoya polynomial of the square of a path and of the square of a cycle. Also, the n-diameter and n-Wiener index of each of the two graphs are determined.
Keyword: n-diameter, n-Hosoya polynomial, n-Wiener index, path square and cycle square.

Introduction:
The n-distance [1] in a connected graph ) E , V ( G  of order t is the minimum distance from a singleton , V v to an (n-1)-subset S , is the sum of the minimum distances of all pairs (v,S) in the graph G , that is: The n-diameter of G is defined by: , then the n-Hosoya polynomial of G is defined by: We can obtain the n-Wiener index of G from the n-Hosoya polynomial of G as follows: For a vertex v of a connected graph G, let k) G, (v, C n be the number of (n-1)-subsets S of vertices of G such that k S) (v, d n  , for 3 n  , . The n-Hosoya polynomial of the vertex v, denoted by ) It is clear that for all 0 k  , For more information about these concepts, see the References [1,2,5,6].
The next lemma will be used in proving our results.

Lemma 1.1:[1]
Let v be any vertex of a connected graph G. If there are r vertices of distance 1 k  from v , and there are s vertices of distance more than k from v , then, . Notice that the square of complete graph, star graph, wheel graph, complete bipartite graph are complete graphs.
In [1,2,3,4] , the n-Hosoya polynomials for many special graphs and many compound graphs are obtained . In this paper , we continue such works by obtaining the n-Hosoya polynomials of the square of paths and cycles.

The n-Hosoya Polynomial of the Square of a Path:
In this section , we obtained the n-Hosoya polynomial of the square 2 t P of a path t P of order t . We shall consider two main cases of 2 t P according to the parity of t.    (1). Let t be even , then , for all 2 n  .
(2). Let t be odd , then 1 Thus , if n is odd , then , If n is even , then ,

Theorem 2.2:
For any 3 n  , the n-Hosoya polynomial of 2 t P , 6 t  , is given by: From Fig.2.2, we notice that in 2 t P , there are two vertices of degree 2, two vertices of degree 3, and 4 t  vertices of degree 4. Thus, using formula (1.4.5) in [1], we obtain (2.2.2). For each vertex w and given k , let So, by Lemma 1.1, we have , for k , ...
So, using Lemma 1.1, we obtain , for So , using Lemma 1.1 , we get , Therefore, using Lemma 1.1, we get , for k r , ...

The n-Hosoya Polynomial of the Square of a Cycle :
There are many classes of connected graphs G in which for each k, ; such graphs are called [2] vertex-ndistance regular graphs, and for the given value of n , t n 2 , where v is any vertex of G and t is the order of G .
For 2 r  , 2 t C is redrawn in Fig. 3 To find the n-eccentricity of 1 v , we partition