Wiener Index for Generalized Hierarchical Product

Graphs

Jayalakshmi G1, Saravanan D2, Vaishnavi R3

1,2 Department of Mathematics, Karpaga Vinayaga College of Engineering and Technology, Chengalpattu.

3 Department of Mathematics, Vidhya Sagar Arts and Science College, Chengalpattu.

Abstract: A topological representation of molecule is called molecular graphs. A molecular graph is a collection of points

representing the atoms in the molecule and set of lines representing the covalent bonds. These points are named vertices and the lines

are named edges in graph theory language. The Wiener index is one of the oldest and most widely used Topological indices.

Originally the wiener index is defined as the sum of the distances between any two carbon atoms in an alkane, interms of carbon-

carbon bonds and is also called path number. This wiener index was motivated by various mathematical properties and chemical

applications. In this paper we study wiener index for generalized hierarchical product of graphs.

Index Terms- Molecular Graph, Wiener Index

1. INTRODUCTION

In order to obtain the structure – activity relationships in which theoretical and computational methods are based it is necessary to find

appropriate representations of the molecular structure of chemical compounds. These representations are realized through the

molecular descriptors. Molecular descriptors are numbers containing structural information derived from the structural representation

used for molecules under study. One of such kind of molecular can be carried out through molecular graph. The descriptors are

numerical values associated with chemical constitution for correlation of chemical structure with various physical properties, chemical

reactivity or biological activity.

They are derived from a topological representation of molecules and can be considered as structure-explicit descriptors in contrast

with those structure cryptic descriptors, such as quantum chemical ones and structure implicit as hydrophobicity and electronic

constants.

The main paradigm of medicinal chemistry is that biological activity, as well as physical, physiochemical and chemical properties of

compound depends on their molecular structure. Based on this paradigm of chemical Crum Brown and Fraser published the first

quantitative structure activity relationship in 1868. This paradigm is guiding the discovery of new lead compounds. A Lead

compound is any chemical compound that shows the biological activity. A lead compound is not a drug but approach to search it.

The role of topological indices in drug discovery is analyzed and updated taking into account the most recent advances in lead

discovery strategies, such as drug design, virtual screening, combinatorial library design and similarity and dissimilarity database

searching.

At this point in our narrative, we pause to gain the high ground concerning the development of chemical graph theory as a whole.

Following the major turbulence of the 1850s and 1860s, generated by the advent of structure and valence theories, the need for

appropriate formalisms for the further development of chemistry became clear. Graph theory represented a very natural formalism for

chemistry became clear.

Graph theory represented very natural guises. Over the succeeding years the use of graph theory in chemistry was assume an

increasingly explicit form. The applications began to multiply so fast that chemical graph theory bifurcated in manifold ways to

evolve into an assortment of different specialism. In general, it may be commented that the development of chemical graph theory has

not been a particularly smooth one. Indeed, it can be said, that periods of great interest in the field have been followed by almost total

neglect. The episodic nature of the interest by chemists has reflected to a large extent the themes of current chemical interest at the

time. Thus, in the 1930s, many chemists were involved in the synthesis of a whole range of new types of molecules, and determining

the number of possible structures which could be made theoretically became of some significance. It was during the 1930s that isomer

enumeration studies had their heyday.

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Chemical bonding theory had been a topic of perennial interest, though the topological nature of simple bonding theory began to be

realized only in the late 1950s. As a consequence, in the period immediately following, an explosion occurred in the number of papers

treating the subject of bonding theory from a graph theoretical standpoint. In every recent time, in the 1970s, chemical graph theory

seems to have become broadly fashionable again, even among a fair number of mathematicians. The current fissionability of our

subject appears to date from the 1960s, when Balaban began publishing an important series of papers bearing the general title

‘Chemical Graphs’ One measure of the interest in a subject is the number of papers published in the field. If we apply this test to

chemical graph theory, it would seem that Balaban’s pioneering work sparked a growing avalanche of interest. Future reviews of the

field are likely to focus on specialized themes rather than attempt the near impossible task of embracing the whole area in a single

survey. A two-volume monograph by Trinajstic[7,12,16] offers an excellent introduction to the subject.

2. TOPOLOGICAL REPRESENTATION OF MOLECULES

A representation of an object giving information only about the number of elements composing it and their connectivity is named as

topological representation of an object. A topological representation of a molecule is called molecular graph. A molecular graph is a

collection of points representing the atoms in the molecule and set of lines representing the covalent bonds. These points are named

vertices and the lines are named edges in graph theory languages. Some important definitions on the molecular graph are the

following.

Hydrogen-depleted graph: A molecular graph which hydrogen atoms are not considered. Valency of atom or vertex degree 𝑖 (𝑖): The number of bonds incident in 𝑖.

Edge or bond degree of 𝒆𝒌 (𝒆𝒌): The number of bonds adjacent to 𝑒𝑘 Two bonds are adjacent if they are incident to the same

atom. The following relation is maintained

(𝑒𝑘) = (𝑣𝑖) + (𝑣𝑗) – 2, 𝑣𝑖 𝑎𝑛𝑑𝑣𝑗 and 𝑒𝑘

Topological distance between atoms 𝒊 and j : is the shortest path between both atoms.

Topological distance dij : is the length of the shortest path between vertices i and j.

Adjacency Matrix A: is a square and symmetric matrix of order n whose elements aij are ones or zeros if the corresponding vertices i

and j are adjacent or not.

Bond Matrix: is a square and symmetric matrix of order m whose elements eij are ones or zeros if the corresponding bonds i and j are

adjacent or not.

Distance matrix D: is a square and symmetric of order n whose elements dij correspond to the topological distances between atoms i

and j.

Spectral moments of a matrix: are the traces, i.e., the sum of the main diagonal, of the different powers of the corresponding matrix.

A topological index is the numerical result of any graph invariant. They are numbers calculate from a graph representing a molecule,

which does not depend on the numbering of the graph vertices or edges.

Topological indices may be used as simple numerical descriptors in a comparison with physical, chemical, or biological parameters of

molecules in quantitative structure- property relationship (QSPR) and in quantitative structure – activity relationship (QSPR). The

most of the proposed topological indices are related either to a vertex adjacency relationship (atom – atom connectivity) in the graph

(molecular structure) G or topological distances in G. Therefore, the origin of topological indices can be traced either to the adjacency

matrix of a graph or to the distance matrix of a graph.

Topological indices are also classified according to their nature in first, second and third generation.

First generation topological indices are based on integer graph properties, such as topological distances. The most representative

indices of this class are Wiener index W, plat index F, Hosoya index and the Centrix index of Balaban B. and C. From these

topological indices the only that has been used in drug discovery research is the Wiener index.

Second generation Topological indices are real numbers based on integer graph properties. Most of the TIS used in drug discovery

today are of this class. The most successful set of such molecular descriptors are the molecular connectivity indices and it introduced

by Randic.

Third generation Topological indices are those real numbers based on real number local properties of the molecular graph. These

indices are of recent introduction, have very low degeneracy and offer possibility of a wide selection. Other third generation

Topological indices are based on information theory applied to terms of distance sums or newly introduced non symmetrical matrices

and these Topological indices are not having any application in drug discovery research.

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3. WIENER INDEX

The Wiener index is one of the oldest and most widely used topological indices in the quantitative structure – property relationship

(QSPR) originally the wiener index is defined as the sum of the distances between any two carbon atoms in an alkane, in terms of

carbon – carbon bonds is called the path number. Wiener also suggested a simple method for the calculation of the path number.

Multiply the number of carbon atoms on the side of any bond by those on other side, W is the sum of the products. In the intial

applications, the Wiener index is employed to predict physical parameters such as boiling points, heats information, heats of

vaporization, molar volumes, and molar refractions of alkances by the of simple QSPR models.

Drugs and other chemical compounds are often modeled as polygonal shapes, where each vertex represents an atom of the molecule,

and covalent bonds between atoms are represented by edges between the corresponding vertices. This polygonal shape derived from a

chemical compound is often called its Molecular graph, and can be a path, a tree or in general a graph. An indicator defined over this

molecular graph, the Wiener index, has been shown to be strongly correlated to various chemical properties of the compound.

W is the sum of the products. In the initial applications, the Wiener index is employed to predict physical parameters such as boiling

points, heats information, heats of vaporization, and molar volumes

Other third generation Topological indices are based on information theory applied to terms of distance sums or newly introduced non

symmetrical matrices and these Topological indices are not having any application in drug discovery research.

Given an undirected graph, denote by the length of the shortest path between two distinct vertices 𝑣𝑖 , 𝑣𝑗 𝑉. The Wiener index [3,5,6],

𝑊 (𝐺) is defined as

2

1)( GW i

j

)( , ji vvd

Consider the graph G with vertices 𝑣1, 𝑣2, 𝑣3, 𝑣4labeled in figure 3.1

Figure 3.1

Here

1),(,1),(,1)(,2),(,1),( 43423,23121 vvdvvdvvdVvdvvd

Therefore

𝑊(𝐺) = 8111221),(

ji

ji

vvd

𝑊(𝐺) = (1 + 2 + 2 + 1 + 1 + 1 + 2 + 1 + 1 + 2 + 1 + 1) = 8

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Wiener index for a bi regular caterpillar graph is

Figure 3.2

Chemical compound: (2,3,4) dimethy1 is pentane

Figure 3.3

3.1 The Wiener Theorem: If 𝐺 is a connected graph, then its Wiener index,𝑊(𝐺), is – by definition – equal to the sum of distances

between all pairs of vertices of 𝐺. Details on this graph Invariant, and on its chemical applications, can be found in some of the many

surveys [4, 5, 7, 11, 12]. Let 𝑇 be a tree (= acyclic connected graph), and let 𝐸(𝑇) be its edge set. Then the Wiener theorem can be

stated as follows:

Let T be a tree on n vertices. Then,

𝑊(𝑇) = ∑ 𝑁2(𝑇 − 𝑣) + (𝑛

2)

𝑒∈𝑉(𝑇)

3.2 The Doyle–Graver formula: In connection with Theorem 2, a formula discovered by Doyle and Graver [6] deserves to be

mentioned. Using the same notation as in Introduction, denote by 𝑁3(𝐹) the sum over all triplets of components, of the product of the

number of vertices of three components of F,

𝑁3(𝐹) = ∑ 𝑛(𝑇𝑖)

1≤𝑖<𝑗<𝑘≤𝑝

𝑛(𝑇𝑗)𝑛(𝑇𝑘)

If p = 1 or p = 2, then 𝑁3(𝐹) = 0. If p = 3 and p = 4, then

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𝑁3(𝐹) = 𝑛(𝑇1). 𝑛(𝑇2). 𝑛(𝑇3)

And

𝑁3(𝐹) = 𝑛(𝑇1). 𝑛(𝑇2). 𝑛(𝑇3) + 𝑛(𝑇1). 𝑛(𝑇2). 𝑛(𝑇4) + 𝑛(𝑇1). 𝑛(𝑇3). 𝑛(𝑇4) + 𝑛(𝑇2). 𝑛(𝑇3). 𝑛(𝑇4)

respectively, etc.

3.3 Vertex Wiener theorem for general graphs:

In order to extend Theorem 2 to general (connected) graphs, one needs to take into account that such graphs may possess several

shortest paths connecting the same pair of vertices. In addition, G − v needs not be disconnected.

The betweenness centrality B(x) of a vertex x ∈ V (G) is the sum of the fraction of all-pairs shortest paths that pass through x:

𝐵(𝑥) = ∑𝜎𝑢,𝑣(𝑥)

𝜎𝑢,𝑣𝑢,𝑣∈𝑉(𝐺)⧵{𝑥}

𝑢≠𝑣

where 𝜎𝑢,𝑣 denotes the total number of shortest (𝑢, 𝑣) −paths in G and 𝜎𝑢,𝑣(𝑥) represents the number of shortest (𝑢, 𝑣) −paths passing

through the vertex x. It is one of the most important centrality indices and it was introduced by Anthonisse [1], and popularized by

Freeman [8] (see also [2]).

Now, we state the extension of Theorem 2. Notice before that for a vertex v in a tree T, it holds 𝐵(𝑣) = 𝑁2(𝑇 − 𝑣).

3.4 Mycielski Construction:

From a simple graph 𝐺, Mycielski’s construction produces a simple graph 𝑀(𝐺) containing 𝐺

Start with G having vertex set {𝑣1, 𝑣2, . . . , 𝑣𝑛}

Add vertices 𝑈 = {𝑢1, 𝑢2 . . . . . . . . 𝑢𝑛} and one more vertex 𝑊

Add edges to make 𝑢, adjacent to all 𝑁𝐺 (𝑉𝑖)

Finally let 𝑁(𝑤) = 𝑈

One iteration of Mycielski’s construction from the graph C8, Where 𝐶𝑛, is a cycle of length 𝑛, yields the graph shown in

figure

In fact 𝑘-chromatic triangle free graph 𝐺, the Mycielski’s construction produces a (𝑘 + 𝑙) −chromatic triangle-free graph 𝑀(𝐺).

Figure 3.4

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4. WIENER INDEX FOR GENERALIZED HIERARCHICAL PRODUCT OF GRAPHS

4.1 Hierarchical Product of Graph:

Let 𝐺𝑖 = (𝑉𝑖 , 𝐸𝑖) be n graphs with a distinguished or root vertex, labeled 0. The hierarchical product 𝐺1 ⊓ 𝐺2 ⊓ ……⊓ 𝐺𝑛 is the graph

with vertex set 𝑉1 × 𝑉2 × 𝑉3 × . . . . . . . . 𝑋. . . . . 𝑉𝑛 and the edges are defined as [1]

(𝑔1, 𝑔2, ………𝑔𝑛)~

{

(𝑔1

′, 𝑔2, ………𝑔𝑛) 𝑖𝑓𝑔1~𝑔1′ 𝑖𝑛 𝐺1,

(𝑔1, 𝑔2′, ………𝑔𝑛) 𝑖𝑓𝑔2~𝑔2

′ 𝑖𝑛 𝐺2 𝑔1 = 0

(𝑔1, 𝑔2, 𝑔3′………𝑔𝑛) 𝑖𝑓𝑔3~𝑔3

′ 𝑖𝑛 𝐺3 𝑔1 = 𝑔2 = 0

(𝑔1, 𝑔2, ………𝑔𝑛′) 𝑖𝑓𝑔𝑛~𝑔𝑛

′ 𝑖𝑛 𝐺𝑛 𝑔1 = 𝑔2 = ⋯𝑔𝑛 = 0

Fig 4.1 The hierarchical product 𝑷𝟒 ⊓ 𝑷𝟑 ⊓ 𝑷𝟐

4.2 Generalized Hierarchical Product of Graph:

Given n graphs 𝐺𝑖 = (𝑉𝑖 , 𝐸𝑖) and non-empty vertex subsets 𝑈𝑖𝑉𝑖 , 𝑖 = 1,2, . . . 𝑛 − 1, the generalized hierarchical product

𝐺1(𝑈1) ⊓ 𝐺2(𝑈2) ⊓. . . .⊓ 𝐺𝑛−1(𝑈𝑛−1) ⊓ 𝐺𝑛 𝑖s the graph with the vertex set 𝑉1 × 𝑉2 × 𝑉3 × . . . . . . . . 𝑋. . . . . 𝑉𝑛 and the adjacency is

given by(2)

(𝑔1, 𝑔2, ………𝑔𝑛)~

{

(𝑔1

′, 𝑔2, ………𝑔𝑛) 𝑖𝑓𝑔1~𝑔1′ 𝑖𝑛 𝐺1,

(𝑔1, 𝑔2′, ………𝑔𝑛) 𝑖𝑓𝑔2~𝑔2

′ 𝑖𝑛 𝐺2 𝑔1 ∈ 𝑈1(𝑔1, 𝑔2, 𝑔3

′………𝑔𝑛) 𝑖𝑓𝑔3~𝑔3′ 𝑖𝑛 𝐺3 𝑔𝑖 ∈ 𝑈𝑖𝑖 = 1,2

(𝑔1, 𝑔2, ………𝑔𝑛′) 𝑖𝑓𝑔𝑛~𝑔𝑛

′ 𝑖𝑛 𝐺𝑛 𝑔𝑖 ∈ 𝑈𝑖𝑖 = 1,2, . . 𝑛 − 1

Note that is all the subsets 𝑈𝑖 are single tons (That is, the trivial graph with only one vertex), then the resulting is the hierarchical

product. In the case 𝑛 = 2, let 𝐺 and 𝐻 be two graphs and 𝜑 ≠ 𝑈 ⊆ 𝑉(𝐺) Then the adjacency in 𝐺(𝑈) ⊓ 𝐻 is given by

(𝑔, ℎ) ~ (𝑔’, ℎ’) {

𝑔 = 𝑔’ ∈ U and h = h′in H

orh = h ∈ U′ and 𝑔~𝑔’in G

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Figure shows the generalized hierarchical product of 𝑃𝑛(𝑈) ⊓ 𝑃2 with 𝑉(𝑃𝑛) = {1,2, . . . . , 𝑛} and 𝑈 = {1,3, . . . . , 𝑛}, where n is odd.

Figure 4.2

5. EXTENSION OF GENERALIZED HIERARCHICAL PRODUCT OF GRAPHS AND CONCLUSION

5.1 FOUR NEW SUMS OF GRAPHS AND THEIR WIENER INDICES

Let 𝐺 be a connected graph.

(a) 𝑆(𝐺) is obtained from 𝐺 by replacing each edge of 𝐺 by a path of length two.

(b) 𝑅(𝐺) is obtained from 𝐺 by adding a new vertex corresponding to each edge of 𝐺, then joining each new vertex to the end

vertices of the corresponding edge.

(c) 𝑄(𝐺) is obtained from 𝐺 by inserting a new vertex into each edge of 𝐺, then joining with edges those pairs of new vertices on

adjacent edges of 𝐺.

(d) 𝑇(𝐺) Has as its vertices the edges and vertices of 𝐺. Adjacency in 𝑇(𝐺) is defined as adjacency or incidence for the

corresponding elements of 𝐺. (This graph is called the total graph of 𝐺. (e) The line graph of 𝐺, denoted 𝐿(𝐺), has the edges of 𝐺 as vertices with two vertices in 𝐿(𝐺) adjacent if, as edges of 𝐺, they

have an endpoint in common.

Figure 5.1

Let 𝐹 be one of the symbols 𝑆, 𝑅, 𝑄 𝑜𝑟 𝑇. The 𝐹 -sum 𝐺1+𝐹𝐺2 is a graph with the set of vertices 𝑉(𝐺1+𝐹𝐺2) = (𝑉(𝐺1)𝑈𝐸(𝐺1)) ×

𝑉(𝐺2) and two vertices (𝑔1, 𝑔2) and (𝑔1’, 𝑔2’) of 𝐺1+𝐹𝐺2 are adjacent if and only if [ 𝑔1 = 𝑔1’ and 𝑔2~ 𝑔2’ in 𝐺2] or [𝑔2 = 𝑔2’ and

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𝑔1~𝑔1’ in 𝐹(𝐺1)]. Note that 𝑈 = 𝑉(𝐺) 𝑉(𝐹(𝐺)) ,then 𝐺1+𝐹𝐻 = 𝐹(𝐺)(𝑈) ⊓ 𝐻. Thus we have a new and short method in

computing the wiener index of 𝐺 +𝐹𝐻

Let 𝐺 = (𝑉1, 𝐸1) and 𝐻 = (𝑉2, 𝐸2) be two connected graphs. Suppose that 𝑈 = 𝑉(𝐺) 𝑉(𝐹(𝑈)). Then if 𝐹 = 𝑄 𝑜𝑟 𝑇, then the

Wiener index of graph 𝐺 +𝐹𝐻 is equal to:

|𝑉2|2(𝐹 (𝐺)) + (|𝑉1| + |𝐸1|)2𝑊(𝐻) +

2

1(|𝐸1|

2 + |(𝐸1|)(|𝑉2|2 − |𝑉2|)

Also if F = S or R, then the Wiener index of graph G +FH is equal to

|𝑉2|2𝑊(𝐹(𝐺)) + (|𝑉1| + |𝐸1|)2𝑊(𝐻) + |𝐸1|𝑉2(|𝑉2| − 1).

Figure 5.2

6. CONCLUSION

During the past 20 years, the QSPR and QSAR technique have gained wide acceptance in physical, organic, analytical, medical

chemistry, bio chemistry, toxicology and environmental science In the recent years a modification of the Wiener number was

developed by the elements of the matrices of vertex and edge – weighted molecular graphs. The Wiener number and its modification

are widely used to account for isometric variations of many physic chemical properties such as boiling points, molar volumes of

alkenes and so on. In this paper, we have studied the Wiener Index and Zagreb index for Mycielski’s construction, generalized

product and four new sums of graphs.

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