Schultz and Modified Schultz Polynomials for Vertex – Identification Chain and Ring – for Hexagon Graphs

بلل ةقلحو ةلدلدل سأ ر قباطتل ةلدعملا زتلوشو زتلوش دودح تاددعتم ةيساددلا تاناي  أ يلع محمد دمح الله دبع نيدم دومحم  تايضايرلاو بهساحلا مهلع ةيلك ،تايضايرلا مسق  قا رعلا ،لصهملا ،لصهملا ةعماج  ت ا :ثحبلا ملاتسا خير  90 / 94 / 0909          ت ا :ثحبلا لوبق خير  90 / 90 / 0909  صخلملا  داجيا هه ثحبلا اذه نم فدهلا  أ ر قباطتل ةلدعملا زتلهشو زتلهش دودح تاددعتم تاقلحلل ةقلحو ةلسلسل س ةيسادسلا  اضيأ امك ،  .امهليلدو ةلدعملا زتلهشو زتلهش ليلد اندجو  ةيحاتفملا تاملكلا : ةلدعملا زتلهش ،زتلهش ،  ةقلحو ةلسلسل سوؤر قباطت .


INTRODUCTION:
We will let all graphs in this paper to be connected, finite, undirected and simple, which means empty from loops and multiple edges. Let be a connected simple graph, and and denote the sets of vertices and edges, respectively, of .
In any graph represent the number of vertices the order of and denoted that by symbol | |, and we called the number of edges the size of , and denoted that by symbol | | We say for any two vertices in adjacent in if there exists edge between them, and we write , as well as we say the edge incident on and . We called the degree of vertex as the number of edges incident on it and denoted that by as such that for vertex in [5]. Now, we define the distance between any two vertices in . The distance is the length of a shortest path that join between and in which is denoted by or . We called the maximum distance between any two vertices and in the diameter and denoted that by [4]. In 2005, Gutman introduced the graph polynomials related to the Schultz and modified Schultz indices [12], and in 2011, Behmaram, et al. found the Schultz polynomials of some graph operation [3]. Farahani [9], gave Schultz and modified Schultz polynomials of some Harary graphs in 2013. Ahmed and Haitham studied Schultz and modified Schultz polynomials, indices, and index average for two Gutman's operations [1]. Also they found general formulas for Schultz and modified Schultz polynomials, indices, and index average of cog-special graphs [2]. Also there are many studies about their applications ( [6,7,8,10,11]). Schultz had introduced and studied in 1989 Schultz index (molecular topological index) [18]. Then, in 1997 Klavžar and Gutman introduced the modified Schultz index [17]. They have defined Schultz and modified Schultz, indices, respectively, as: ∑ . ∑ . Schultz and modified Schultz polynomials are considered very important polynomials through studying some properties of their coefficients. Schultz and modified Schultz polynomials are defined, respectively, as: We can obtain the indices of Schultz and modified Schultz by taking derivative of them with respect to at , as explained below. | and | .
While we can obtain the average of the Schultz and modified Schultz indices for connected graph with order that are defined as: and .
In any connected graph , we refer to the set of unordered pairs of vertices which are distance apart by the symbol and let | | . Now let that be the set of all unordered pairs of vertices in , which are of distance and of It is obvious that ∑ | | where | |. Finally, Schultz indices are considered very interesting to determine some properties of chemical structures, see more ( [13,14,15,16]).

The Vertex -Identification Chain (VIC) -Graphs:
Let be a set of pairwise disjoint graphs with vertices then the vertex-identification chain graph of with respect to the vertices is the graph obtained from the graphs by identifying the vertex with the vertex for all (See Fig. 2-1) in which: . Some Properties of Graph : ( ) ∑ . The equality of both bounds are satisfied at complete graphs, but the upper bound is satisfied at path graphs in which , are end-vertices of for .

If
, for all , where is a connected graph of order , we denoted by .

P4.
If when , then, we have two subsets of : P5. If when , 3, then, we have two subset of : P6. If when , , then, we have three subsets of : Thus | | for , .
P7. If then, we have two subsets of : P8. If then | | because: P9. If then | | because: From P 1 to P 10 and

The Vertex -Identification Ring (VIR) -Graph:
Let be a set of pairwise disjoint graphs with vertices then the vertex-identification Ring graph of with respect to the vertices is the graph obtained from the graphs by identifying the vertex with the vertex for all (See Fig. 2-2) where .
The equality of both bounds are satisfied at complete graphs but the upper bound is satisfied at path graphs in which , are end-vertices of for . If , for all , where is a connected graph of order , we denoted by .