Hosoya Polynomials of The Width Distance of Some Cog-Special Graphs

Tahir H. Ismail Tahir_hs@yahoo.com Ahmed M. Ali ahmedgraph@uomosul.edu.iq College of Computer Sciences and Mathematics University of Mosul, Mosul, Iraq Received on: 29/03/2011 Accepted on: 21/06/2011 ABSTRACT In this paper, we give some definitions and properties of the width distance and find the Hosoya polynomials, Wiener indices, and the average distances of some special cog-graphs with respect to the width distance. Keyword: Width distance, Hosoya polynomial, Wiener index, average distance, coggraphs. ةصاخلا ةننسملا تانايبلا ضعبل ةيضرعلا ةفاسملل ايوسوه دودح تاددعتم  ليعامسإ نسح رهاط   يلع دمحم دمحأ  تايضايرلاو بوساحلا مولع ةيلك ،  لصوملا ةعماج  خيرات  ملاتسا  ثحبلا : 29 / 03 / 2011   لوبق خيرات  ثحبلا : 21 / 06 / 2011  صخلملا  ايوييسوا و يي ييعام ايي اض سييم ةيييضرعلا ةفاييإملل ةاوييللاو ايماييعالا اييعي اايياعض بييا عييحتلا الييا هييف .ةصاللا ةننإملا تانايبلا اعتل ةيضرعلا ةفاإملل ةفاإملا ل عمو رنيو ليل و  لا ةيحاتفملا تاملك :  .ةننإم تانايب ،ةفاإملا ل عم ، رنيو ليل ، ايوسوا و عام ، ةيضرعلا ةفاإملا


Introduction:
Because of the development of parallel architectures for interconnection computer networks, there has recently been interest in generalizations of the distance concept, for examples, the n-distance [1], Steiner distance [5], and width distance [2].
In [2] A. S. Aziz obtained Hosoya polynomials of the width distance of some special graphs and some compound graphs. It would be interesting to compute this polynomial for various graphs and to study its properties. It would also be interesting to see if this object yields any useful information in chemistry , group theory , or computer science. For more information about these concepts and their relation to networks , see the article of Hsu [6].
Let u and v be any two distinct vertices in a connected graph G, define the container C(u,v) as a set of vertex-disjoint paths between vertices u and v, i.e., any two paths in C(u,v) have only u and v in common. The container width w = w(C(u,v)), is the number of paths in the container, i.e., w(C(u,v)) = | C(u.v) |. The length of a container l = l(C (u,v)) is the length of the longest path in C(u,v). For a fixed positive integer w, define the width distance (w-distance) between u and v ( [2], [7]) as: where the minimum is taken over all containers C(u,v) of width w. We may denote the width distance between any two distinct vertices u and v in G by if there is no confusion.
Note that when w = 1, then ) v , u ( d w  reduces to the usual distance between vertices u and v which denoted by ) v , u ( d ([4], [7]). Therefore, in this paper, we take 0 k w 2   , in which 0 k is the connectivity of G [3].
To define the diameter and radius of any connected graph G with respect to the width distance , we first define the eccentricity. The w-eccentricity ) v ( e w  of the vertex v in a graph G is the greatest possible wdistance from v to all other vertices of G, that is The w-diameter The minimum w-distance The w-radius , a graph G may contain more than one w-centeral vertex ,we define the w-center of a graph G, , as the set of all w-central vertices of G.
It is clear that the w-distance in connected graphs does not satisfy the metric axioms, because for is not satisfactory for all vertices u and v of G. To show that, we take the next example .
Example: Let G be a graph of order 10 and size 12, as shown in Fig.1.1.   Fig. 1.1.
Concerning the width minimum distance ) G ( m 2 , we have = iff G contains a triangle or a cycle of order 4. y u v

Hosoya Polynomial of The Width Distance
Let G be a connected graph of order p, size q, and let w be a fixed positive integer such that 0 k w 2   , and w-diameter is   w . The Wiener index of the wdistance (w-Wiener index) is the sum of all w-distances in G, that is [2] The Hosoya polynomial with respect to the width distance function * w d (called w-Hosoya polynomial) is defined by [2]: is the number of unordered pairs of distinct vertices that has k w-distance apart, The average w-distance, , of G is defined as : We define the w-Hosoya polynomial of a vertex v in G as : is the number of vertices in V(G)-{v} that are at w-distance k from vertex v. We note that From the previous discussion, we note that Let S be a nonempty subset of V(G), then define Moreover, we define ) k , G , S ( C * w as the number of pairs {u,v} of vertices of S such that Finally, if is a partition of V(G) , then

On The Coefficients of The w-Hosoya Polynomial
is the w-Hosoya polynomial of a connected graph G for a fixed w, then the sequence of a graph G invariant , for a fixed w , , often possesses some distinguished properties.
and it is strong-unimodal if (3.1) holds without equalities. For example, [2] : If is the graph as shown in Fig. 3.1, then and it is j-semi-palindromic if (4.2) holds for k j ) For example, [2] : If p C is a cycle of order p, 3 p  , then is palindromic if p is odd , and it is 1-semi-palindromic if p is even.
3. The polynomial ) For example, [2] : If 2 m P is the square of a path m P of order 5 m  , then is monotonically decreasing. Definition: A cog-complete graph c m K is the graph constructed from a complete graph , and 2 k 0 = . Thus, w must be 1 or 2 only. Therefore, we take w = 2.
We notice that Proof:Let u and v be any two distinct vertices of , then we consider the following cases: To find 2-Hosoya polynomial of c m K , 4 m  , there are three cases: If u and v are adjacent to a common vertex of V, then there are two internally disjoint paths between the vertices u and v of lengths 2 and 3; therefore Hence , the number of such pairs of vertices is m. (ii). If u and v are not adjacent to a common vertex of V, then there are two internally disjoint paths between the vertices u and v, each has length 3, therefore Hence, the number of such pairs of vertices is Case (3): If U u  , and V v , then (i). There are two internally disjoint paths between the vertices u and v of lengths 1 and 2, when Hence, the number of such pairs of vertices is 2m. (ii). There are two internally disjoint paths between the vertices u and v each of length 2, when ) Hence, the number of such pairs of vertices is From three cases, we obtain Corollary 4.1.2: Remark: , and

Cog-star Graphs:
Definition:  It is clear that then

Case (2):
, then there are three internally disjoint paths between v and v having lengths 2 , Thus, Hence, the number of pairs of {u,v}, , is 2(m-1), except when m is even and Case ( Hence , , for all 5 m  .

Cog-wheel Graphs
; and for 1 m j , having lengths 4 , and ; and for ). Hence,

Case (2):
 , then we consider 3 subcases: , then there are four internally disjoint paths between v and v having lengths 1, 2, 2 , m-2, , then there are three internally disjoint paths between the vertices v and v , having lengths 2, , then the number of pairs { v , v } , , is (m-1), except when m is odd and 2 , then the number of pairs { v , v } is 2 , and the number of these pairs is m-1.
, then the two paths have length 1, and 2, therefore , and u and v are neither adjacent nor adjacent to a common vertex of , then the two paths have lengths 3, and , then the number of pairs { u , u } such that is m, except if m is even, in which the number of { u , u } such that Hence , from this case and (