LISTA INICIAL DEL PRODUCTO

*Head, Department of Mathematics, Srimad Andavan Arts & Science College (Autonomous), Trichy–05. **Asst.Professor, Department of Mathematics, Srimad Andavan Arts & Science College (Autonomous), Trichy – 05

E-mail ID: [email protected], [email protected]

ABSTRACT :

The Wiener index of a graph is defined as the sum of distances between all pairs of vertices in a connected graph. Wiener index correlates well with many Physio Chemical properties of organic compounds and as such has been well studied over the last quarter of a century. In this paper we prove some general results on Wiener Lower and Upper Sum for graphs. Keywords: Knr - Tree, Wiener Lower Sum of Knr – Tree

MSC Code: 05CXX

1. INTRODUCTION

Chemical Graph theory is used to model physical properties of molecules called alkanes. Indices based on the graphical structure of the alkanes are defined and used to model both the boiling point and melting point of the molecules. Alkanes are organic compounds exclusively composed of carbon and hydrogen atoms. Molecular descriptors are “Terms that characterize a specific aspect of a molecule”. Topological indices have been defined as those “Numerical values associated with chemical contribution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity”. 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 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.

Given the structure of an organic compound, the corresponding (molecular) graph is obtained by replacing the atoms by vertices and covalent bonds by edges (double and triple bonds also correspond to single edges unless specified otherwise).The Wiener index is one of the oldest molecular-graph based structure-descriptors, first proposed by the Chemist Harold Wiener as an aid to determining the boiling point of paraffins. The study of Wiener index is one of the current areas of research in Mathematical Chemistry. There are good correlations between Wiener index of molecular graphs and the Physico-Chemical properties of the underlying organic compounds.

2. COMPLETE GRAPH

A Complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. The complete graph with n vertices is denoted by Kn.

Let Kn be complete graph with n vertices, let V(Kn) = {u1, u2, u3, . . . un} and E(Kn) = {uiui+1/1 ≤ i < n} {unu1} {uiuj / 1 ≤ i < n and i < j ≤ n} be respectively vertex set and edge set of Kn.

Example - 1

Figure – (i) Complete K9 Example – 2

 

Figure – (ii) Complete graph Kn 

Given Kn is a complete graph, let Knr graph be obtained by appending r pendant edges, 1 ≤ r ≤ n first edge to any one vertex, second edge to any remaining one vertex and third edge to any where remaining one vertex and so on

Let Knr be a complete graph with n vertices and r pendent vertices, let V(Knr) = {u1, u2, . . . un, v1, v2, v3, . . .vr} and E(Kn) = {uiui+1/1 ≤ i < n} ∪ {unu1} ∪ {ui uj / 1 ≤ i < n and i < j ≤ n} ∪ {upv1, uqv2, . . . ukvr /1 ≤ r ≤ n & 1 ≤ p, q,. . . k ≤ n} be respectively the vertex set and the edge set of Knr.

Example – 3

Figure – (iii) Kn r

3. WIENER LOWER SUM WL(G) 3.1 Definition

Let G = (V(G), E(G)) be a connected undirected graph, any two vertices u, v of V(G), (u, v) denotes the minimum distance between u and v. Then the Wiener Lower Sum WL_{(G) of the graph is defined by}

WL_{(G) =}

∑

(

)

∈ ( ) , , 2 1 G V v u v u

δ _{where (u, v) = min d(u,v)}

3.2 Theorem

The Wiener Lower Sum of a Knr – Graph is WL(Knr) =

(

)

(

)

2 1 2 2 4 1 ₊ 2 _{+ −} − r r n n n Proof:

Let Kn be a complete graph with n vertices then the Wiener Lower Sum of Kn is WL_(K n) =

(

)

2 1 − n n .

Adding one pendant edge unv1 to the end vertex then the Wiener Lower Sum is WL_(K

n1) = WL(Kn) + 

∑

_i δ

(

v1 ,ui

)

, where 1 ≤ i ≤ n, (v1,ui) – minimum distance between vertices v1 and ui

WL_(K

n1) = WL(Kn) + 2(n-1) + 1

Again adding another pendant edge un-1v2 to the last but one end vertex then the Wiener Lower Sum is WL_(P

n2) = WL(Pn1) +

∑

_i δ

(

v1 ,ui

)

+

∑

_i δ

(

v2 ,ui

)

, where 1 ≤ i ≤ n, (v2, ui) – minimum distance between vertex v2 to all the vertices of

Kn1 WL(Kn2) = WL(Pn1) + 2(n – 1) + 1 + 3

Also, again adding one more pendant edge un-2v3 to the last but two end vertices then the Wiener Lower Sum is WL_(K

n3) = WL(Kn2) + ∑ δ(v3, ui) + ∑ δ(v3, vk)

where 1 ≤ i ≤ n and 1 ≤ k < 3, δ(v3,ui) – Minimum distance between vertex v3 to all the vertices of Kn2 WL_(K

n3) = WL(Kn2) + 2(n – 1) + 1 + 3 + 3

Continue this process of adding pendent edge until 1 < r < n then the Wiener Lower Sum of Knr is WL_(K

nr) = WL(Knr-1)+∑ δ(vr, ui)+∑ δ(vr, vk) where 1 ≤ i ≤ n, and 1 ≤ k < r, Distance between vertex vr to all the vertices of Pnr-1 WL_(K nr) = WL(Knr-1) + 2(n – 1) + 1 + ∑ (2+3+. . . r terms) + ∑ (3+4+. . . r terms) + . . .   =n

(

n₂−1

)

+2(n–1) + 1 + . . . ‘r’ terms WL(Knr) =

(

)

(

)

2 1 2 2 4 1 ₊ 2 _{+ −} − r r n n n 3.3 Programming in C #include<stdio.h> #include<conio.h> void main() { int i,j,k,n,s=0,r; clrscr();

printf("\nNo of Vertices in a graph - \t"); scanf("%d",&n);

printf("\nNo of Pendent Vertices - \t"); scanf("%d",&r); for(i=1;i<n;i++) { for(j=i;j<n;j++) s=s+1; }

printf("\nWiener Lower Sum is WL(Kn) = \t%d",s); for(i=1;i<=r;i++)

{

for(j=1;j<n;j++) s=s+2;

for(i=2;i<=r;i++) { for(j=1;j<i;j++) s=s+3; } s=s+r;

printf("\nWiener Lower Sum is WL(Knr) = \t%d",s); getch();

} Output

No of Vertices in a graph - 8

No of Pendent Vertices - 5

Wiener Lower Sum is WL(Kn) = 28 Wiener Lower Sum is WL(Knr) = 133 4. CONCLUSION

Hence the Wiener Lower Sum of Knr - Tree are useful for systems like radar tracking, remote control, communication networks and radio-astronomy etc. Estimation of Wiener Lower Sum for other graph or tree is under investigation.

As it is well known that the valence of the carbon atom is four and vertex with more than degree four is not agreeable in reality, it is the interest of the chemist to use these work and study the stability of the compound and existence of the molecule with the properties with which we have analyzed the wiener index. We have taken only the concept of the wiener index and explored for general graphs with certain properties.

REFERENCES

[1] H. Wiener, Structural Determination of Paraffin Boiling Points, J. Amer. Chem. Soc. 69 (1947), 17-20 [2] I.Gutman, O.Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin, 1986

[3] L. Xn and X. Guo, Catacondensed Hexagonal Systems with Large Wiener Numbers, MATCH Commun. Math. Comput. Chem. 55 (2006), 137-158

[4] R. Balakrishnan & K. Ranganathan, “A Text book of Graph Theory”, Springer, New York, 2000

[5] A.A. Dobrynin, R. Entringer, and I. Gutman, Wiener Index of Trees: Theory and Applications, Acta appl. Math. 66 (2001), 211-249

[6] A.A. Dobrynin, I. Gutman, S. Klavzar and P. Zigert, Wiener Index of Hexagonal Systems, Acta appl. Math. 72 (2002), 247-294

[7] E. Estrada and E. Uriarte, Recent Advances on the Role of Topological Indices in drug Discovery Research, Cur.Med.Chem. 8 (2001), 1573-1588

[8] M. Miller and J. Siran, Moore Graphs and Beyond: A Survey of the Degree/Diameter Problem, The Elec. J. Combin. DS14 (2005), 1-61

[9] Bruce E. Sagan, Yeong-Nan Yeh, “The Wiener Polynomial of a Graph”, MI 48824-1027, U.S.A.

[10] K. Thilakam & A. Sumathi, “Wiener Index of Chain Graph”. Aryabhatta J. of Mathematics & Informatics Vol. 5 (2) (2013) pp. 347-352.

[11] G. Nirmala & M. Muyugan : “Characteristics of Peterson graph in cycle Matrix with algebraic graph Theory” Aryabhatta J. of Mathematics’ & Info. Vol. 6 (2) (2014) 335-342.

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