Restricted Detour Polynomial of a Straight Chain of Wheel Graphs

Abstract


INTRODUCTION
In this research paper, we consider finite simple connected graphs. For undefined concepts and notations on the theory of graphs, we refer the reader to [5]. Topological indices are graph invariants that play important roles for studying and analyzing the physicochemical properties of molecules. Some of the most worthy types of topological indices of graphs are distance-based topological indices, degree-based topological indices, and spectrumbased topological indices. The Wiener index is the first studied topological index. For more details on the concept of topological indices of graphs, we refer the reader to [1,4,6,7,8,11,12].
The concept of restricted detour distance was first proposed in 1993 by Chartrand, Johns and Tian [6]. For standard graph theory and acquired the restricted detour polynomial and restricted detour index of certain graphs see [2,3,9,10]. Specifically, let and be two distinct vertices in a connected graph . The (standard) distance ( ) from a vertex to a vertex in a graph is the smallest length of a path in [5]. An induced path of length ( ) is called a restricted detour path. The restricted detour distance between two vertices and of a graph is the length of a longest path for the induced condition 〈 ( )〉 and indicated ( ). The restricted detour polynomial depends on restricted detour distance and is denoted by ( ) and defined by ( ) ∑ ( ) * + , where the summation is taken over all unordered pairs ( ) of a distinct vertices and of . The index is also based on the restricted detour distance and denoted by ( ) and is defined by , where the summation is taken over all unordered pairs of vertices of [10]. Also, ( ) ( ) . A wheel graph of order , is a graph that contains a cycle of order , and for which every vertex in the cycle is connected to one other vertex which is known as the hub (or the center).
In 2012, Ali and Gashaw [1] obtained the restricted detour polynomials of chain a hexagonal ladder graph. In 2017, Ali, I. D. and Herish computed the restricted detour polynomial of edge-Identification of two wheel graphs [3]. Ali, I. D, obtained restricted detour polynomial of some cycle related graphs [2]. In this study, we obtained the restricted detour polynomial and restricted detour index of chain k-wheels consisting of one row Let us indicate the j-th copy of in ( ) by , for . Then the two consecutive wheels and will have the same proportion of the common edge ( ( ) ( ) ) ( ( ) ( ) ). Now, for ; and if k is even, then the two consecutive wheels and will share the common edge ( ( ) ( ) ) (  Consequently, we reach the following reduction formula in which ( ) is the polynomial corresponding to all possibilities of for which and .
Now, after some calculation having carried out, we readily

Restricted Detour Polynomial of Straight
Chain of Wheels ( ), for and In this section, we will discover the restricted detour polynomials and the restricted detour indices of straight chains of wheels ( ) for and .
The restricted detour polynomial of ( ) for is given in the next two propositions.
Proposition 3.1. For odd , the restricted detour polynomial of ( ) is given by Proof. We refer to Figure2, with , and denote Then the polynomial in this case is given by ), then the path between is is a longest path of length ( ). Then the polynomial in this case is given by ( ) ( ) .
), then the path between is Proof. We can proof the proposition by using similar techniques and steps followed in the proof of Proposition 3.1.
The restricted detour polynomial of ( ) for is given in the next two proposition.

If
, then the restricted detour path Next, we obtain the restricted detour index of ( ), by taking the derivative of ( ( ) ) at obtained from Propositions 3.1. and 3.2. .

Proof.
(1) Taking the derivative of ( ( ) ) given in Proposition 3.2., at we get where . Now, simplifying the results above, we get ( ( )) as given in the statement of the proposition.
The restricted detour index of ( ) is obtained in the next corollary by taking the derivative of ( ( ) ) at obtained in Propositions 3.3. and 3.4. (2) For odd 7, we have ( ( )) (

Chains of Wheels ( ) for
In this section, we will discover the restricted detour polynomials and the restricted detour indices of the straight chain of wheels ( ), where .
In the next theorem, we shall find the restricted detour polynomial of ( ) odd and (for even and odd ). ), then the restricted detour path is is a longest path of length .
is a longest path of length . The polynomial in this case is given by ( ) .
(2) ( ) ( ( ) ) for , then the restricted detour path is ), for , . The polynomial in this case is given by Similarly if is even.
This completes the proof. In the next two theorems, we shall find the restricted detour polynomial of ( ) even and .
Theorem 4.2. For even and odd , the restricted detour polynomial of ( ) is given by Proof. We can proof the theorem by using similar techniques and steps followed in the proof of Theorem 4.1.
is the longest path of length .
Then the polynomial is ( ) ) then the path between is   Proof. Obvious.