INTRODUCTION:
The introduction of Wiener index and Wiener polarity
index by Harry Wiener in his paper titled “Structural Determination of Paraffin
Boiling Points” opened scope of studying and determining physical properties
from graph structures [1]. He stated that the boiling points of organic
compounds as well as their physical properties depend functionally upon the
number kind and structural arrangement of atoms in a molecule. He further
stated that within a group of isomers both the number and the kind of atoms are
constant and variations in physical properties are due to change in structural
inter-relationships alone. Further in this paper, he has developed a
relationship between the boiling point, Wiener index and Wiener polarity index
given by the equation 
t = (98/n2) w + 5.5p, Where t is the boiling point of group of isomers. w = w0 – w, where w0 = (1/6) (n-1) (n) (n+1), n-
vertices and w is the path number i.e., the sum of the distances between any
two carbon atoms in a molecule, in terms of carbon-carbon bonds. p = p0- p, where p0 = n-3, p the polarity number
defined as the number of pairs of carbon atoms which are separated by three carbon-carbon bonds. He has provided a list of the detailed results obtained by applying this equation to the 37 paraffins from C4H10 to C8H18. He further extended this method for the boiling point data available for the nonanes and decanes. After Wiener numerous researchers had developed various techniques to determine physical and chemical properties from topological indices. In [2] boiling point model from the topological indices is developed and compared with the existing models. In [3] the boiling point is predicted with the weighted sum of the generalized Zagreb index, the Second Zagreb index, Wiener index for vertex weighted graphs. In [4] a property relationship study for prediction of boiling point of aliphatic alkenes is determined. In [5] a formula for computing the Wiener index in terms of Binary Hamming labeling of G is determined. In [6] a method of determining Wiener index of trees from its subtree is determined.
To calculate the Wiener index of a graph with n vertices, a n×n distance matrix is required. This means that distance between n2 pair of vertices should be determined. A reduction in this calculation would reduce the manual difficulty involved in physical property determination. Eccentricity is a graph property involving distance. Eccentricity is determined only for vertices. This means that for a chemical graph with n vertices, only n calculations are required. So physical properties can be determined from eccentricities only. Then a enormous manpower require for distance calculation may reduce. In this paper we propose method of determining the boiling point of alkanes and alkenes is obtained.
PRELIMINARIES:
In this section we provide the basic details of Wiener index and chemical graph that is required in the proposed calculation.
Molecules:
A molecule is an electrically neutral group of two or more atoms held together by chemical bonds [ 7 ]. Molecules are distinguished from ions by their lack of electrical charge. However, in quantum physics, organic chemistry, and biochemistry, the term molecule is often used less strictly, also being applied to polyatomic ions. In the kinetic theory of gases, the term molecule is often used for any gaseous particle regardless of its composition. According to this definition, noble gas atoms are considered molecules as they are in fact mono atomic molecules. (Snapshot 1) shows the example of a molecule- Sucrose molecule [ 8 ].
Snapshot 1: Sucrose molecule
Molecular structure:
The arrangement of chemical bonds between atoms in a molecule (or in an ion or radical with multiple atoms), specifically which atoms are chemically bonded to what other atoms with what kind of chemical bond, together with any information on the geometric shape of the molecule needed to uniquely identify the type of molecule. (Snapshot 2) gives an example of the molecular structure of a molecule [ 9 ].
Snapshot 2: Molecular Structure
Isomeric group:
An isomer is a molecule with the same molecular formula as another molecule, but with a different chemical structure. (Snapshot 3) provides the structural formulas for an isomer [10]
Snapshot 3: Structural formulas for C4H10O isomers
Boiling Points:
The boiling point of a substance is the temperature at which the substance boils, or enters a state of rapid evaporation. For pure water this is 100° Celsius or 212° Fahrenheit. This is measured at one atmosphere, that is, the air pressure at sea level. The boiling point of a liquid depends on the pressure of the surrounding air [11].
|
Compound |
Boiling Point (°C) |
Compound |
Boiling Point (°C) |
|
Benzyl Alcohol |
205 |
Ethyl Benzote |
213 |
|
Glycerol |
290 |
Methyl salicylate |
223 |
|
Ethylene Glycol |
197 |
Nitrobenzene |
211 |
|
Phenol |
182 |
Aniline |
184 |
|
o-Cresol |
191 |
o-Toluidine |
200 |
|
Benzaldehyde |
178 |
Chlorobenzene |
132 |
|
Acetophenone |
202 |
Bromobenzene |
156 |
|
Phenyl Acetate |
196 |
Benzoyl Chlorid |
197 |
Snapshot 4: Boiling point of different compounds
Alkanes:
In organic chemistry, an alkane, or paraffin (a historical name that also has other meanings), is an acyclic saturated hydrocarbon. In other words, an alkane consists of hydrogen and carbon atoms arranged in a tree structure in which all the carbon-carbon bonds are single. Alkanes have the general chemical formula CnH2n+2 [12]. (Snapshot 5) gives the examples of Alkanes [13].
Snapshot 5: Example for Alkanes and its structure
Alkenes:
In organic chemistry, [14] an alkene is an unsaturated hydrocarbon that contains at least one carbon–carbon double bond. The words alkene and olefin are often used interchangeably (see nomenclature section below). Acyclic alkenes, with only one double bond and no other functional groups, known as mono-enes, form a homologous series of hydrocarbons with the general formula CnH2n. (Snapshot 6) shows the examples of Alkene molecules [15].
Snapshot 6: Example for Alkenes and its structure
Graph:
A graph G is an ordered
triple V(G), E(G), x(G) consists of non-empty set of vertices
called V(G), a non-empty set of edges called E(G), and an incident function 
(G)
which associates with each edges of G and unordered pair of vertices. (Figure
1) shows the example of a graph [16].
Fig. 1: Example of a Graph
Path:
A path in a graph is a finite or infinite sequence of edges which connect a sequence of vertices which are all distinct from one another [ 17 ]. Examples of paths with n = 1, 2, 3, 4 vertices are shown in (Figure 2).
Fig. 2: Examples for Paths
Distance:
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance [ 18 ]. Notice that there may be more than one shortest path between two vertices.[ 19 ] If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite. The following example (Figure 3) shows the distance of two vertices is 5.
Fig. 3: A Graph showing distance between two vertices is 5.{\displaystyle d(u,v)}{\displaystyle d(v,u)}
Eccentricity:
The eccentricity of any vertex v in a graph G is defined as the length of the longest possible from the vertex to any other vertex in the graph.
Fig. 4: A Graph with eccentricity 4, 3, 4, 4, 3
For example for the graph in (Figure 4) the length of the longest possible path is to vertex 5 namely 1 to 2 to 3 to 4 to 5. So the eccentricity of vertex 1 is 4. Similarly the eccentricity of the vertices 2, 3, 4 and 5 are 3, 4, 4, 3 respectively.
Wiener index:
In chemical graph theory, the Wiener index (also Wiener number) is a topological index of a molecule, defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph representing the non hydrogen atoms in the molecule [ 6 ]. The distances dij form the so-called distance matrix D(G) = [ dij] of the graph G. The Wiener index of G is the number
W(G) = 
= ½
Wiener index from distance matrix
Fig. 5: Distance matrix of Tree T
The Wiener index of a path with n – vertices is n (n2-1) / 6. [ 20 ]
PROPOSED METHOD OF BOILING POINT DETERMINATION:
The boiling point of organic compounds give importance to many physical properties and structural characteristics.
Harry Wiener in his paper on the structural determination of paraffin boiling point establish that the boiling point can be determined based on the structural arrangement of atoms in a molecule. He has given a relation that boiling points of paraffins are given by the linear formula TB = aW + b P + c where a,b and c are constants for a given isomeric groups while P and W are structural variables determined by Wiener and later labeled as Wiener index and Wiener polarity index respectively. The regular procedure to determine Wiener index uses distance between every pair of vertices in any chemical structure ( here after we shall refer to them as chemical graphs or simply graphs ).
Eccentricity for every vertex is defined as the distance of the longest possible path from the vertex. In a distance matrix, the distance between all pair of vertices is required, while to determine the eccentricity of any vertex only longest possible path is required. If a method can be determined to find the boiling point using eccentricity then lot of manual calculation gets reduced. Specifically as the size of graph increases, we attempt to determine the boiling point using eccentricity values only.
An equation of the form B = α W β relating the boiling point B and Wiener index W is approximated by B = 181 W 0.1775 [ 21 ] . We attempt to determine similar formula using eccentricity of any given vertex. The chemical graph structure 2,2 – dimethyl pentane is seen in (Figure 6).
Fig. 6: Chemical graph structure of 2, 2- dimethyl pentane
The eccentricity of the vertices is represented as label of vertex. (Figure 7) shows the eccentricity of above Fig 6 in which the sum of the eccentricity = 24.
Fig. 7: A graph showing the eccentricities of Fig. 6
We shall define the eccentricity of any graph G to be the sum of eccentricities of all the vertices in G. The graph in (Figure 8) displays a graph of the boiling point eccentricity index data.