Hosoya polynomial and wiener index of Abid-Waheed graph ( AW ) a 8 and ( AW ) a 10

Abstract

Molecular structures are characterised by the Hosoya polynomial and Wiener index, ideas from mathematical chemistry and graph theory. The graph representation of a chemical compound that has atoms as vertices and chemical bonds as edges is called a molecular graph, and the Hosoya polynomial is a polynomial related to this graph. As a graph attribute that remains unchanged under graph isomorphism, the Hosoya polynomial is known as a graph invariant. It offers details regarding the quantity of distinct non-empty subgraphs within a specified graph. A topological metric called the Wiener index is employed to measure the branching complexity and size of a molecular graph. For every pair of vertices in a molecular network, the Wiener index is the total of those distances. In this paper, discussed the Hosoya polynomial, Wiener index and Hyper-Wiener index of the Abid-Waheed graphs (AW)_a⁸ and (AW)_a¹⁰. This graph is similar to Jahangir’s graph. Further, we have extended the research work on the applications of the described graphs.

Keywords

Wiener index Abid-Waheed Hosoya polynomial diameter distance connected graph

1 Introduction

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Vertices (also known as nodes or points) make up a graph in this context, which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the major objects of study in discrete mathematics. In physical, biological, social, and information systems, graphs can be used to model a wide variety of linkages and processes. Many practical problems can be represented by graphs. Cybernetics, a branch of computer science, uses graphs to describe computational devices, communication networks, data structure, computation flow, etc. For instance, a directed graph, in which the vertices represent web pages and the directed edges indicate links from one page to another, can be used to depict the link structure of a website. a similar strategy. In chemistry, a graph’s vertices and edges serve as a natural representation of atoms and bonds, respectively, in a molecule. This method is particularly useful when processing molecular structures on computers, including chemical editors and database searches. In statistical physics, graphs can depict the dynamics of a physical process on such systems as well as the local connections between interacting system components. Similar to this, in computational neuroscience, graphs can be used to represent the connections between brain regions that function together to produce a variety of cognitive processes. In these graphs, the vertices correspond to the various regions of the brain, and the edges correspond to the connections between them. Graph theory plays an important role in the electrical modelling of electrical networks; here, weights are associated with the resistance of the wire. The interested readers may read [3 , 25]. The chemical graph theory is more important because of isomorphism in chemical structures. Using the isomorphism in chemical structures, many applications can be found in medicine and food production [8 , 23]. Topological indices (TIs) are quantitative measurements of molecular networks that are unaffected by graph isomorphism. A vast range of topological indices has been produced, and a lot of work has been spent on computing the indices of various molecular graphs and networks [2 , 11]. Among the many vertex degree-based topological indices, the first and second Zagreb indices of a graph are among the oldest, most well-known, and most thoroughly investigated. In an article published in 1972 [13], Gutman and Trinajstic introduced these indices in order to investigate the structure-dependency of the total π-electron energy. The reformulated first Zagreb index of a few chemical graphs was calculated by Nilanjan De [18] as an application of generalised hierarchies made from graphs. A few degree-based indices of networks derived from honeycomb networks were obtained by Wei Gao et al. [26]. Shao et al. [21] study degree-based indices for the bismuth tri-iodide chain and sheet. To get around the laborious process of computing a particular type of indices for a certain category of graphs, many algebraic polynomials are found in the literature. For instance, the Hosoya polynomial, whose differentiation at one produces the Wiener and Hyper-Wiener indices, is crucial in the field of distance-based topological indices. The M-polynomial plays a key role in computing the degree-based topological indices. It was introduced by Deutsch and Klavzar [10] where its role to computing degree-based indices was shown to be parallel to the role of the Hosoya polynomial for distance-based indices. Many readers may have read [4, 24]. Following that, many M-polynomial-related works were completed. Munir et al. [16] and computed M-polynomial and degree-based topological indices for nanostar dendrimers. When chemist Harold Wiener proposed the Wiener index while calculating paraffin’s boiling temperature in 1947 [27], the use of TIs in biology and chemistry followed. He showed correlations between the molecular graph index and the physical and chemical characteristics of organic compounds. Wiener used the index for trees and examined its usage for correlations between the physicochemical properties of alkanes, alcohols, amines, and their analogous compounds [17, 20].

The Jahangir graph and the generalisation of the Abid-Waheed graph are identical [6 , 22]. Deo [9] studied the applications of graph theory in computer science. Chakraborty et al. [7] studied the neutrosophic single-valued number and its applications in networking problems and group decision-making problems. Meenakshi et al. [15] studied the application of the neutrosophic optimal network using the operation of graphs. This form of graph’s pattern has application to computer networking; hence, it is used to evaluate both the effectiveness and efficiency of the network. Since the Wiener index is a distance-based index, it provides information about the graph’s location. The Wiener index, also known as the Wiener number, is a topological index of a molecule used in chemical graph theory. It is calculated as the sum of the shortest paths between all pairs of vertices in the chemical graph that represent the non-hydrogen atoms in the molecule. The Wiener index of a graph G^* is defined as $W (G^{*}) = \frac{1}{2} \sum_{(r, s) \in V (G^{*})} φ (r, s)$ In 1988, H. Hosoya introduced the Hosoya polynomial of a graph, a counting polynomial that counts the number of distances between pathways in the graph of various lengths. Hosoya introduced the polynomial

H (G^*, k) = ∑_{(r,s)∈V(G^*)}φ (r, s) k^φ(r,s) where φ (r, s)is the distance between r and s. Obtain the Wiener index from the first-time differentiation of the Hosoya polynomial at k=1. The Wiener index of the Hosoya polynomial is defined as $W (G^{*}) = {{\frac{\partial H (G^{*}, k)}{\partial k}} |}_{k = 1}$ First derivative of the Hosoya polynomial is defined as $W (G^{*}) = (H)^{'} (G^{*}, k) = {{\frac{\partial H (G^{*}, k)}{\partial k}} |}_{k = 1}$ Second derivative of the Hosoya polynomial is defined as $W (G^{*}) = (H)^{″} (G^{*}, k) = {{\frac{\partial^{2} H (G^{*}, k)}{\partial k}} |}_{k = 1}$ Hyper-Wiener index is defined as $HW (G^{*}) = (H)^{'} (G^{*}, k) + \frac{1}{2} (H)^{″} (G^{*}, k)$ Abid-Waheed et al. [1] introduced the Abid-Waheed graph (AW)_a^b, where a ≥ 1 and b ≥ 3, and use the Hosoya polynomial to find the Wiener index of (AW)_p⁶. The aim of this paper is to continue the studies of the classes of (AW)_p⁸ and (AW)_p¹⁰, and find the Wiener index, Hyper-Wiener index of this class of graphs using the Hosoya polynomial.

2 Methodology

The Abid-Waheed graph is represented by (AW)_a^b, where a ≥ 1 and b ≥ 3, which consists of ab+1 vertices and a(b+1) edges. A graph produced bya-cycles (each of order b), each meeting at an external vertex of degree a, is shown in Figure 1.

This section’s primary goal is to calculate the values of the Hosoya polynomial and Wiener index and Wiener-index for the Abid-Waheed graph’s (AW)_a⁸and (AW)_a¹⁰. First compute the Hosoya polynomial by determining the various distances and their cardinality in a graph, and then we use this polynomial to get the Wiener index for (AW)_a⁸and (AW)_a¹⁰. The φ (x, y) symbol represents the number of unordered pairs of nodes, and the distance φ (x, y) is represented by φ (G^*, s). The topological diameter of a graph G^* is referred to as D(G^*). s may have a maximum value equal to D(G^*), but it lies between 1 to D(G^*).

Example (i) The Hosoya polynomial H (G^*, k)of G^*=(AW)_a⁸is determined as follows: $H (G^{*}, k) = \sum_{s = 1}^{10} φ (G^{*}, s) k^{s}$ $H (G^{*}, k) = {\begin{matrix} φ (G^{*}, 1) k + φ (G^{*}, 2) k^{2} \\ + φ (G^{*}, 3) k^{3} + φ (G^{*}, 4) k^{4} \\ + φ (G^{*}, 5) k^{5} + φ (G^{*}, 6) k^{6} \\ + φ (G^{*}, 7) k^{7} + φ (G^{*}, 8) k^{8} \\ + φ (G^{*}, 9) k^{9} + φ (G^{*}, 10) k^{10} \end{matrix}$ The diameter of (AW) ₃ ⁸ is 10. The values of φ (G^*, s), where s lies between 1 to 10, that is 1 ≤ s ≤ 10 are given in Table 1, which shows the distances along with the pertinent frequencies.

Table 1
Cardinality of φ (G^, s) for each pair of vertices of (AW)₃⁸ where 1 ≤ s ≤ 10

φ (G^, s) Cardinality φ (G^, s)* Cardinality

φ (G^, 1)* 27 φ (G^, 6)* 42

φ (G^, 2)* 33 φ (G^, 7)* 36

φ (G^, 3)* 42 φ (G^, 8)* 24

φ (G^, 4)* 42 φ (G^, 9)* 12

φ (G^, 5)* 39 φ (G^, 10)* 3

φ (G^, s)*	Cardinality	φ (G^, s)*	Cardinality
φ (G^, 1)*	27	φ (G^, 6)*	42
φ (G^, 2)*	33	φ (G^, 7)*	36
φ (G^, 3)*	42	φ (G^, 8)*	24
φ (G^, 4)*	42	φ (G^, 9)*	12
φ (G^, 5)*	39	φ (G^, 10)*	3

The Wiener index of G^*= (AW)₃⁸ is obtained from the first derivative of H (G^*, k) at k = 1

$H ({AW}_{3}^{8}, k) = {\begin{matrix} 27 k + 33 k^{2} + 42 k^{3} + 42 k^{4} \\ + 39 k^{5} + 42 k^{6} + 36 k^{7} \\ + 24 k^{8} + 12 k^{9} + 3 k^{10} \end{matrix}$

$W ({AW}_{3}^{8}) = {\begin{matrix} 27 + 66 k \\ + 126 k^{2} + 168 k^{3} \\ + 195 k^{4} + 252 k^{5} \\ + 252 k^{6} + 192 k^{7} \\ + 108 k^{8} + 30 k^{9} \end{matrix}} at k = 1$

$W ({AW}_{3}^{8}) = 1416$

Example(ii) The Hosoya polynomial H (G^*, k)of G^*=(AW)₃¹⁰ is determined as follows: $\begin{matrix} H (G^{*}, k) & = \sum_{s = 1}^{12} φ (G^{*}, s) k^{s} \\ H (G^{*}, k) & = {\begin{matrix} φ (G^{*}, 1) k + φ (G^{*}, 2) k^{2} \\ + φ (G^{*}, 3) k^{3} + φ (G^{*}, 4) k^{4} \\ + φ (G^{*}, 5) k^{5} + φ (G^{*}, 6) k^{6} \\ + φ (G^{*}, 7) k^{7} + φ (G^{*}, 8) k^{8} \\ + φ (G^{*}, 9) k^{9} + φ (G^{*}, 10) k^{10} \\ + φ (G^{*}, 11) k^{11} + φ (G^{*}, 12) k^{12} \end{matrix} \end{matrix}$ The diameter of (AW) ₃ ¹⁰ is 12. The values of φ (G^*, s), where s lies between 1 to 12, that is 1 ≤ s ≤ 12 are given in Table 2, which shows the distances along with the pertinent frequencies.

Table 2

Cardinality of φ (G^*, s) for each pair of vertices of (AW)₃¹⁰ where 1 ≤ s ≤ 12

φ (G^, s)*	Cardinality	φ (G^, s)*	Cardinality
φ (G^, 1)*	33	φ (G^, 7)*	54
φ (G^, 2)*	39	φ (G^, 8)*	48
φ (G^, 3)*	48	φ (G^, 9)*	48
φ (G^, 4)*	60	φ (G^, 10)*	24
φ (G^, 5)*	66	φ (G^, 11)*	12
φ (G^, 6)*	51	φ (G^, 12)*	3

The Wiener index of G^*= (AW)₃¹⁰ is obtained from the first derivative of H (G^*, k)

at k = 1 $\begin{matrix} H ({AW}_{3}^{10}, k) & = {\begin{matrix} 33 k + 39 k^{2} + 48 k^{3} \\ + 60 k^{4} + 66 k^{5} + 51 k^{6} \\ + 54 k^{7} + 48 k^{8} + 48 k^{9} \\ + 24 k^{10} + 12 k^{11} + 3 k^{12} \end{matrix} \\ W ({AW}_{3}^{10}) & = {\begin{matrix} 33 + 78 k \\ + 144 k^{2} + 240 k^{3} \\ + 330 k^{4} + 306 k^{5} \\ + 378 k^{6} + 384 k^{7} \\ + 432 k^{8} + 240 k^{9} \\ + 132 k^{10} + 36 k^{11} \end{matrix}} at k = 1 \\ W ({AW}_{3}^{10}) & = 2733 \end{matrix}$

3 Results and discussion

Theorem 3.1. Let G^*= (AW)_a⁸ be a simple connected graph. Where a ≥ 1, then the Hosoya polynomial is $\begin{matrix} H (G^{*}, k) = {\begin{matrix} 9 ak + (\frac{a^{2} + 19 a}{2}) k^{2} \\ + (2 a^{2} + 8 a) k^{3} + (4 a^{2} + 2 a) k^{4} \\ + (6 a^{2} - 5 a) k^{5} + (7 a^{2} - 7 a) k^{6} \\ + (6 a^{2} - 6 a) k^{7} + (4 a^{2} - 4 a) k^{8} \\ + (2 a^{2} - 2 a) k^{9} + (\frac{a^{2} - a}{2}) k^{10} \end{matrix} \end{matrix}$

Table 3
Cardinality of φ (G^, s) for each pair of vertices of (AW)_a⁸ where 1 ≤ s ≤ 10

φ (G^, s) Cardinality φ (G^, s)* Cardinality

φ (G^, 1)* 9a φ (G^, 6)* 7a² - 7a

φ (G^, 2)* $\frac{a^{2} + 19 a}{2}$ φ (G^, 7)* 6a² - 6a

φ (G^, 3)* 2a² + 8a φ (G^, 8)* 4a² - 4a

φ (G^, 4)* 4a² + 2a φ (G^, 9)* 2a² - 2a

φ (G^, 5)* 6a² - 5a φ (G^, 10)* $\frac{a^{2} - a}{2}$

φ (G^, s)*	Cardinality	φ (G^, s)*	Cardinality
φ (G^, 1)*	9a	φ (G^, 6)*	7a² - 7a
φ (G^, 2)*	$\frac{a^{2} + 19 a}{2}$	φ (G^, 7)*	6a² - 6a
φ (G^, 3)*	2a² + 8a	φ (G^, 8)*	4a² - 4a
φ (G^, 4)*	4a² + 2a	φ (G^, 9)*	2a² - 2a
φ (G^, 5)*	6a² - 5a	φ (G^, 10)*	$\frac{a^{2} - a}{2}$

Proof. Let G^*= (AW)_a⁸, where a ≥ 1 is a simple connected graph. The graph G^*consists of 8a+1 vertices and 9a links which are shown in Figure 1 and the diameter of G^* is 10.

Fig. 1

Abid -Waheed graph (AW)_a^b.

The Hosoya polynomial H (G^*, k)of a graph G^*whose diameter is 10 is given by $\begin{matrix} H (G^{*}, k) & = \sum_{s = 1}^{10} φ (G^{*}, s) k^{s} \\ H (G^{*}, k) & = {\begin{matrix} φ (G^{*}, 1) k + φ (G^{*}, 2) k^{2} \\ + φ (G^{*}, 3) k^{3} + φ (G^{*}, 4) k^{4} \\ + φ (G^{*}, 5) k^{5} + φ (G^{*}, 6) k^{6} \\ + φ (G^{*}, 7) k^{7} + φ (G^{*}, 8) k^{8} \\ + φ (G^{*}, 9) k^{9} + φ (G^{*}, 10) k^{10} \end{matrix} \end{matrix}$ Using the values present in the above table 3, the Hosoya polynomial of G^* is $\begin{matrix} H (G^{*}, k) = {\begin{matrix} 9 ak + (\frac{a^{2} + 19 a}{2}) k^{2} \\ + (2 a^{2} + 8 a) k^{3} + (4 a^{2} + 2 a) k^{4} \\ + (6 a^{2} - 5 a) k^{5} + (7 a^{2} - 7 a) k^{6} \\ + (6 a^{2} - 6 a) k^{7} + (4 a^{2} - 4 a) k^{8} \\ + (2 a^{2} - 2 a) k^{9} + (\frac{a^{2} - a}{2}) k^{10} \end{matrix} \end{matrix}$

Theorem 3.2. Let G^*= (AW)_a⁸ be a simple connected graph. where a ≥ 1, then Wiener index of G^* is W (G^*) =192a² - 104a.

Proof. The relation between the Hosaya polynomial and the Wiener index of a graph G^* is

$\begin{matrix} W (G^{*}) & = {{\frac{\partial H (G^{*}, k)}{\partial k}} |}_{k = 1} \\ W (G^{*}) & = {\frac{\partial}{\partial k} {\begin{matrix} 9 ak + (\frac{a^{2} + 19 a}{2}) k^{2} \\ + (2 a^{2} + 8 a) k^{3} \\ + (4 a^{2} + 2 a) k^{4} \\ + (6 a^{2} - 5 a) k^{5} \\ + (7 a^{2} - 7 a) k^{6} \\ + (6 a^{2} - 6 a) k^{7} \\ + (4 a^{2} - 4 a) k^{8} \\ + (2 a^{2} - 2 a) k^{9} \\ + (\frac{a^{2} - a}{2}) k^{10} \end{matrix}} |}_{k = 1} \\ W (G^{*}) & = {{\begin{matrix} 9 a + (\frac{a^{2} + 19 a}{2}) 2 k \\ + (2 a^{2} + 8 a) 3 k^{2} \\ + (4 a^{2} + 2 a) 4 k^{3} \\ + (6 a^{2} - 5 a) 5 k^{4} \\ + (7 a^{2} - 7 a) 6 k^{5} \\ + (6 a^{2} - 6 a) 7 k^{6} \\ + (4 a^{2} - 4 a) 8 k^{7} \\ + (2 a^{2} - 2 a) 9 k^{8} \\ + (\frac{a^{2} - a}{2}) 10 k^{9} \end{matrix}} |}_{k = 1} \end{matrix}$ After simplification, we received W (G^*) =192a² - 104a.

Theorem 3.3. Let G^*= (AW)_a¹⁰ be a simple connected graph. Where a ≥ 1, then the Hosoya polynomial is $H (G^{*}, k) = {\begin{matrix} 11 ak + (\frac{a^{2} + 23 a}{2}) k^{2} \\ + (2 a^{2} + 10 a) k^{3} + (4 a^{2} + 8 a) k^{4} \\ + (5 a^{2} + 7 a) k^{5} + (8 a^{2} - 7 a) k^{6} \\ + (9 a^{2} - 9 a) k^{7} + (8 a^{2} - 8 a) k^{8} \\ + (8 a^{2} - 8 a) k^{9} + (4 a^{2} - 4 a) k^{10} \\ + (2 a^{2} - 2 a) k^{11} + (\frac{a^{2} - a}{2}) k^{12} \end{matrix}$

Proof. Let G^*= (AW)_a¹⁰, where a ≥ 1 is a simple connected graph. The graph G^* consists of 10a+1 vertices and 11a links which can be observed from Figure 1 and the diameter of G^* is 12.

Table 4

Cardinality of φ (G^*, s) for each pair of vertices of (AW)_a¹⁰ where 1 ≤ s ≤ 12

φ (G^, s)*	Cardinality	φ (G^, s)*	Cardinality φ (G^, 1)*	11a	φ (G^, 7)*	9a² - 9a
φ (G^, 2)*	$\frac{a^{2} + 23 a}{2}$	φ (G^, 8)*	8a² - 8a
φ (G^, 3)*	2a² + 10a	φ (G^, 9)*	8a² - 8a
φ (G^, 4)*	4a² + 8a	φ (G^, 10)*	4a² - 4a
φ (G^, 5)*	5a² + 7a	φ (G^, 11)*	2a² - 2a
φ (G^, 6)*	8a² - 7a	φ (G^, 12)*	$\frac{a^{2} - a}{2}$

The Hosoya polynomial H (G^*, k)of a graph G^*whose diameter is 12 is given by $\begin{matrix} H (G^{*}, k) & = \sum_{s = 1}^{12} φ (G^{*}, s) k^{s} \\ H (G^{*}, k) & = {\begin{matrix} φ (G^{*}, 1) k + φ (G^{*}, 2) k^{2} \\ + φ (G^{*}, 3) k^{3} + φ (G^{*}, 4) k^{4} \\ + φ (G^{*}, 5) k^{5} + φ (G^{*}, 6) k^{6} \\ + φ (G^{*}, 7) k^{7} + φ (G^{*}, 8) k^{8} \\ + φ (G^{*}, 9) k^{9} + φ (G^{*}, 10) k^{10} \\ + φ (G^{*}, 11) k^{11} + φ (G^{*}, 12) k^{12} \end{matrix} \end{matrix}$ Using the values present in the above table 4, the Hosoya polynomial of G^* is $H (G^{*}, k) = {\begin{matrix} 11 ak + (\frac{a^{2} + 23 a}{2}) k^{2} \\ + (2 a^{2} + 10 a) k^{3} + (4 a^{2} + 8 a) k^{4} \\ + (5 a^{2} + 7 a) k^{5} + (8 a^{2} - 7 a) k^{6} \\ + (9 a^{2} - 9 a) k^{7} + (8 a^{2} - 8 a) k^{8} \\ + (8 a^{2} - 8 a) k^{9} + (4 a^{2} - 4 a) k^{10} \\ + (2 a^{2} - 2 a) k^{11} + (\frac{a^{2} - a}{2}) k^{12} \end{matrix}$

Theorem 3.4. Let G^*= (AW)_a¹⁰ be a simple connected graph. where a ≥ 1, then the Wiener index of G^* is W (G^*) =363a² - 178a

Proof. The relation between the Hosoya polynomial and the Wiener index is W (G^*) =363a² - 178a $W (G^{*}) = \frac{\partial}{\partial k} {{\begin{matrix} 11 ak + (\frac{a^{2} + 23 a}{2}) k^{2} \\ + (2 a^{2} + 10 a) k^{3} \\ + (4 a^{2} + 8 a) k^{4} \\ + (5 a^{2} + 7 a) k^{5} \\ + (8 a^{2} - 7 a) k^{6} \\ + (9 a^{2} - 9 a) k^{7} \\ + (8 a^{2} - 8 a) k^{8} \\ + (8 a^{2} - 8 a) k^{9} \\ + (4 a^{2} - 4 a) k^{10} \\ + (2 a^{2} - 2 a) k^{11} \\ + (\frac{a^{2} - a}{2}) k^{12} \end{matrix}} |}_{k = 1}$ $W (G^{*}) = {{\begin{matrix} 11 a + (\frac{a^{2} + 23 a}{2}) 2 k \\ + (2 a^{2} + 10 a) 3 k^{2} \\ + (4 a^{2} + 8 a) 4 k^{3} \\ + (5 a^{2} + 7 a) 5 k^{4} \\ + (8 a^{2} - 7 a) 6 k^{5} \\ + (9 a^{2} - 9 a) 7 k^{6} \\ + (8 a^{2} - 8 a) 8 k^{7} \\ + (8 a^{2} - 8 a) 9 k^{8} \\ + (4 a^{2} - 4 a) 10 k^{9} \\ + (2 a^{2} - 2 a) 11 k^{10} \\ + (\frac{a^{2} - a}{2}) 12 k^{11} \end{matrix}} |}_{k = 1}$ After simplification, we received W (G^*) =363a² - 178a.

Theorem 3.5. Let G^*= (AW)_a⁸ be a simple connected graph. Where a ≥ 1, then the Hyper- Wiener index of G^* isHW (G^*) =720a² - 546a.

Proof. Let G^*= (AW)_a⁸be a simple connected graph, Where a ≥ 1,then Wiener index of G^* is,

$W (G^{*}) = (H)^{'} (G^{*}, k) = {{\frac{\partial H (G^{*}, k)}{\partial k}} |}_{k = 1}$

By Theorem 3.2. We have,

W (G^*) = (H) ′ (G^*, k) = 192a² - 104a

The second order derivative of the Hosoya polynomial is,

$(H)^{″} (G^{*}, k) = {{\frac{\partial^{2} H (G^{*}, k)}{\partial k}} |}_{k = 1}$

By Theorem 3.2. We have, $(H)^{″} (G^{*}, k) = {{\begin{matrix} (a^{2} + 19 a) \\ + (2 a^{2} + 8 a) 6 k \\ + (4 a^{2} + 2 a) 12 k^{2} \\ + (6 a^{2} - 5 a) 20 k^{3} \\ + (7 a^{2} - 7 a) 30 k^{4} \\ + (6 a^{2} - 6 a) 42 k^{5} \\ + (4 a^{2} - 4 a) 56 k^{6} \\ + (2 a^{2} - 2 a) 72 k^{7} \\ + (\frac{a^{2} - a}{2}) 90 k^{8} \end{matrix}} |}_{k = 1}$ (H) ″ (G^*, k) =1056a² - 884a Hyper-Wiener index is defined as

$HW (G^{*}) = (H)^{'} (G^{*}, k) + \frac{1}{2} (H)^{″} (G^{*}, k)$

$HW (G^{*}) = 192 a^{2} - 104 a + \frac{1}{2} (1056 a^{2} - 884 a)$

HW (G^*) =720a² - 546a .

Theorem 3.6. Let G^*= (AW)_a¹⁰ be a simple connected graph. where a ≥ 1, then the Hyper-Wiener Index of G^* is $HW (G^{*}) = \frac{3175}{2} a^{2} - \frac{2295}{2} a$

Proof. Let G^*= (AW)_a¹⁰, where a ≥ 1 be a simple connected graph, then the Wiener index of the graph is

$W (G^{*}) = (H)^{'} (G^{*}, k) = {{\frac{\partial H (G^{*}, k)}{\partial k}} |}_{k = 1}$ By Theorem 3.4. We have,

W (G^*) = (H) ′ (G^*, k) = 363a² - 178a The second order derivative of the H-polynomial of the graph is,

$(H)^{″} (G^{*}, k) = {{\frac{\partial^{2} H (G^{*}, k)}{\partial k}} |}_{k = 1}$ By Theorem 3.4. We have, $(H)^{″} (G^{*}, k) = {{\begin{matrix} (a^{2} + 23 a) \\ + (2 a^{2} + 10 a) 6 k \\ + (4 a^{2} + 8 a) 12 k^{2} \\ + (5 a^{2} + 7 a) 20 k^{3} \\ + (8 a^{2} - 7 a) 30 k^{4} \\ + (9 a^{2} - 9 a) 42 k^{5} \\ + (8 a^{2} - 8 a) 56 k^{6} \\ + (8 a^{2} - 8 a) 72 k^{7} \\ + (4 a^{2} - 4 a) 90 k^{8} \\ + (2 a^{2} - 2 a) 110 k^{9} \\ + (\frac{a^{2} - a}{2}) 132 k^{10} \end{matrix}} |}_{k = 1}$ (H) ″ (G^*, k) =2449a² - 1939a Hyper-Wiener index is defined as

$HW (G^{*}) = (H)^{'} (G^{*}, k) + \frac{1}{2} (H)^{″} (G^{*}, k)$

$HW (G^{*}) = 363 a^{2} - 178 a + \frac{1}{2} (2449 a^{2} - 1939 a)$

$HW (G^{*}) = \frac{3175}{2} a^{2} - \frac{2295}{2} a$

4 Applications of Abid-Waheed graph (AW)_a^b

The pattern of the Abid-Waheed graph helps us to assign main sources to particular nodes such that the shortest distance between any two nodes is within the minimum. This pattern is useful in the following ways:

(i) Transportation and Arrangements: While taking into account the current traffic situation, GPS navigation systems use shortest path algorithms to determine the fastest path between two points. Robots and autonomous cars plan their moves using shortest path algorithms to minimise impediments and get to their destination as quickly as possible. For characters to move about and navigate in games, pathfinding algorithms that follow the shortest route theory are essential.

(ii) Analysis of Social Networks: In order to optimise supply chain routes and logistics and lower costs and delivery times, shortest path algorithms must be applied. In order to guarantee efficient movement for both passengers and goods, rail and public transport routes are planned by maximising the shortest way.

(iii) Routing on the network: Routers in computer networks employ shortest path algorithms to ascertain the most expedient path for data packets to arrive at their intended destination. Shortest path algorithms are used by telecommunication networks to optimise signal routing, lowering latency and guaranteeing effective communication. Shortest path algorithms can be used to identify the most crucial paths for transmission, which are necessary to analyse contact networks and comprehend the development of illnesses. The shortest path algorithms help reduce travel times and congestion while planning effective public transit routes and designing road networks.

Conclusion

The distinct pattern in the graph structure of the Abid-Waheed graphG^*= (AW) _a ⁶ , where a = 1, 2, 3, ⋯n, is composed of 6 + 1, 2 (6) +1, ⋯ , n (6) +1 the vertices. Each vertex in the degree 3 can be used as a neutral source for a supply chain, or centre. The Abid-Waheed graph’s pattern is hence quite adaptable. The pattern of the Abid-Waheed graph, which is connected to the potency and activity of the active drugs inside the molecular structure, is quite similar to that of dendrimers. The graph G^*= (AW) _a ^b pattern is used to examine the efficacy and activity of the drugs found in similar molecular graph configurations. Network programmes make use of this graph and assess the network’s efficacy.

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Hosoya polynomial and wiener index of Abid-Waheed graph ( AW ) a 8 and ( AW ) a 10

Abstract

Keywords

1 Introduction

2 Methodology

Table 1 Cardinality of φ (G*, s) for each pair of vertices of (AW)38 where 1 ≤ s ≤ 10 φ (G*, s) Cardinality φ (G*, s) Cardinality φ (G*, 1) 27 φ (G*, 6) 42 φ (G*, 2) 33 φ (G*, 7) 36 φ (G*, 3) 42 φ (G*, 8) 24 φ (G*, 4) 42 φ (G*, 9) 12 φ (G*, 5) 39 φ (G*, 10) 3

Conclusion

References

Table 1
Cardinality of φ (G^, s) for each pair of vertices of (AW)₃⁸ where 1 ≤ s ≤ 10

φ (G^, s) Cardinality φ (G^, s)* Cardinality

φ (G^, 1)* 27 φ (G^, 6)* 42

φ (G^, 2)* 33 φ (G^, 7)* 36

φ (G^, 3)* 42 φ (G^, 8)* 24

φ (G^, 4)* 42 φ (G^, 9)* 12

φ (G^, 5)* 39 φ (G^, 10)* 3