The Wiener Index, the Hyper-Wiener Index and the Degree ... - Hikari Ltd

3 downloads 0 Views 447KB Size Report
a molecule. One of the challenges faced by Chemo-informatics is to be able ..... Finally it might be observed that the corona operation allows us to obtain another ... izing the topological structure and properties of Molecular graphs, Research ... [12] V. KBalakrishnan, Graph theory, Schaum's outline, Mcgraw-hill compa-.
Applied Mathematical Sciences, Vol. 8, 2014, no. 85, 4217 - 4226 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43202

The Wiener Index, the Hyper-Wiener Index and the Degree Distance Index of the Corona Cm ◦ Cn Mohamed Essalih LRIT associated unit to CNRST (URAC 29) Faculty of Sciences, Mohammed V-Agdal University, P.O. Box 1014, Rabat, Morocco Mohamed El Marraki LRIT associated unit to CNRST (URAC 29) Faculty of Sciences, Mohammed V-Agdal University, P.O. Box 1014, Rabat, Morocco Abd Errahmane Atmani LRIT associated unit to CNRST (URAC 29) Faculty of Sciences, Mohammed V-Agdal University, P.O. Box 1014, Rabat, Morocco c 2014 M. Essalih, M. El Marraki and A. Atmani. This is an open access Copyright article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract To characterize the topological features of the carbon atom skeleton, or a graph of a hydrocarbon molecule, a number of topological indices have been proposed by chemists. This paper presents some topological indices, like the Wiener index W (Cm ◦ Cn ), the degree distance index DD(Cm ◦ Cn ) and the hyper-Wiener index W W (Cm ◦ Cn ) of the Corona of the cycles Cm and Cn , using the novel methodologies.

Mathematics Subject Classification: 05C12, 05C05 Keywords: Graph, hyper-Wiener index, Wiener index, degree distance index, Corona of two cycles

4218

1

M. Essalih, M. El Marraki and A. Atmani

Introduction

The process of drug discovery is long and tedious. Besides, it is relatively inefficient in terms of hit rate. The identification of candidates through experimental testing is expensive and requires extensive data on the mechanisms of the target protein in order to develop efficient assays. The modeling of the molecular properties and activities can considerably accelerate the process by quickly evaluating large databases of compounds and determining the most likely to bind to a biological activity or physical/chemical property of a molecule. One of the challenges faced by Chemo-informatics is to be able to describe compounds in a simple way, in ordre to use them in similarity studies or to predict their activity, based on information contained in already known compounds. In recent decades, much researches have been carried out to find the best way of representing the information contained in the structure of molecules, and these structures themselves, into a set of real numbers called topological indices; Once these numbers are available, it is possible to establish a relationship there between a molecular property or activity, using the traditional modeling tools. These topological indices realize this is an encoding of chemical information into a vector of real numbers [5]. More than ten million of chemical substances have hitherto been known. Most of them contain carbon atom(s) and are called organic compounds only with a few exceptions (e.g. CO, CO2, HCN). Hydrocarbon molecules composed of carbon (C) and hydrogen (H) atoms and depicted as CmHn (see Figure 2 (a)), play principal role in the whole family of organic compounds. A saturated hydrocarbon is a hydrocarbon that can no longer bind with more hydrogen atoms without destroying its carbon atom skelton. The others are known as unsaturated hydrocarbons, such as benzene (see Figure 1 (a)) carry free electrons, and show interesting electronic and sometimes magnetic properties. The topological structure of a molecule is conventionally expressed by the structural formula composed of the atomic symbols for the components and connecting lines for the chemical bonds, or adjacency relations between the component atoms. In a structural formula, saturated bonds are expressed by single lines (see Figure 2 (c)) while unsaturated bonds are depicted by multiple lines depending on the degree of unsaturation (see Figure 1 (c)). If one substitutes all the atomic symbols and multiple lines of the structural formula into points and single lines, respectively, one gets nothing else except what is called a connected undirected graph defined i n t he g raph t heory. T his g raph m ay be called a molecular graph. Further, in serval cases, all the vertices representing hydrogen atoms and the incident edges are suppressed to give smaller graphs. In the Figure 1 and 2 are given the molecular formulas, structural formulas, and molecular graphs (of the carbon atom skeletons) of saturated and unsaturated hydrocarbon molecules Hexane and benzene, respectively [4][11].

4219

Wiener index, hyper-Wiener index and degree distance index

C6 H6

(a)

H C H

H C

C

C

C

(b)

H

H C H

(C)

Figure 1: Example of an unsaturated hydrocarbons : Benzene

A graph G is a triplet consisting of a vertex set V (G), an edge set E(G), and a relation that associates with each edge two vertices, called its endpoints (see for example Figure 3). We denote |V (G)| = n is the vertex number of G and | E(G)| = m is the edges number of G. A graph which contains neither multiple edges nor loops (the edge uv with u = v) is a simple graph. A named path is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the list (see Figure 2 (c)). A graph G is connected if each pair of the vertices in G belongs to a path. A graph which represents in the plan is called a planar graph. The degree of vertex v in a graph G, which is written deg(v), is the number of edges incident to v, except that each loop at v counts twice, and we call distance between two distinct vertices of graph G, u and v, the smallest length of path between u and v in G.

4220

M. Essalih, M. El Marraki and A. Atmani

The diameter of G, denoted by D(G), is defined as the maximum distance between any two vertices of G, that is, D(G) = max{d(u, v) : ∀(u, v) ∈ V (G)2 }, see [5][12][16] for more details. In the following, we consider only the simple planar connected graphs. Let dG (k) be the number of pairs of vertices of G that are at distance k, λ a

C6 H14

(a)

(b)

H

H

H

H

H

H

H

C

C

C

C

C

C

H

H

H

H

H

H

H

(C) Figure 2: Example of a Hydrocarbon molecules : Hexane

real number, and Wλ (G) = Σk≥1 dG (k)k λ . Wλ (G) is called the Wiener-type invariant of G associated with the real number λ [6][10]. The Wiener index of the graph G equals to the sum of distances between all pairs of vertices of the graph W (G) = Σ{u,v}⊆V (G) d(u, v), and we define the Wiener index of a vertex u in the graph G as w(u, G) = Σv∈V (G) d(u, v) [6][10]. The hyper-Wiener index, is defined as W W (G) = 12 (W1 (G) + W2 (G))[10]. The degree distance index DD(G) of a graph G is defined a s D D(G) = Σ{u,v}⊆V (G) (deg(u) + deg(v))d(u, v ), see [5-7][10][15] for more details. The Corona G1 ◦ G2 of two graphs G1 and G2 (where Gi has ni vertices and mi edges) is defined a s t he g raph G obtained by taking one copy of G1 and n1 copies of G2, and then joining by an edge the ith vertex of G1 to every vertex in the ith copy of G2 (see Figure 4). It follows from the definition of the Corona that G1 ◦ G2 has n1(1 + n2) vertices and m1 + n1(m2 + n2) edges. This fact allows us to recognize that the asso-ciative law never holds G1 ◦ (G2 ◦ G3) and (G1 ◦ G2) ◦ G3 are strictly different graphs. Indeed, the first one has n1(1 + n2 + n2n3) vertices while the second has n1(1 + n2)(1 + n3) vertices, and these numbers are not at all equal, see [1-3][8-9][13-14] [17] for more details.

Wiener index, hyper-Wiener index and degree distance index vn

4221

v1

vn−1 v2

v8

v3

v7

v4

v6

v5

Figure 3: The cycle planar graph C

2

The main Result

Given two cycles Cn1 and Cn2 ; we note by Cn1 ◦ Cn2 the Corona of Cn1 and Cn2 , is defined a s t he g raph o btained by t aking n 1 c opies o f C n2 a nd f or each i, inserting edges between the ith vertex of Cn1 and each vertex of the ith copy of Cn2 for 1 ≤ i ≤ n1 [3][8] (see Figure 4).

Figure 4: The Corona Cn1 ◦ Cn2 of the cycles Cn1 and Cn2

4222

2.1

M. Essalih, M. El Marraki and A. Atmani

The Wiener index of the Corona Cn1 ◦ Cn2

Lemma 2.1. Let Cn1 and Cn2 be the cycles with n1 > 4 and n2 > 4 vertices, respectively. The number of vertices and edges of Cn1 ◦ Cn2 is n = n1 + n1 n2 and m = n1 + 2n1 n2 respectively. And, we have : D(Cn1 ◦ Cn2 ) = n21 + 2 if n1 is even, and D(Cn1 ◦ Cn2 ) = n12−1 + 2 if n1 is odd. Then,

dCn1 ◦Cn2 (k) =

                    

n1 + 2n1 n2 , n1 + 2n1 n2 + n1 (n22−3)n2 , n1 + n1 n22 + 2n1 n2 , n1 + n1 n22 + 2n1 n2 , n1 2 2 + 2n1 n2 + n1 n2 , 2 n1 n2 + n1 n2 , n1 n22 2 , n1 n22 , n1 n22 + 2n1 n2 ,

if if if if if if if if if

k = 1; k = 2; n1 is even, 3 ≤ k ≤ D(Cn1 ◦ Cn2 ) − 3 and D(Cn1 ◦ Cn2 ) > 5 n1 is odd, 3 ≤ k ≤ D(Cn1 ◦ Cn2 ) − 2 and D(Cn1 ◦ Cn2 ) > 4 n1 is even and k = D(Cn1 ◦ Cn2 ) − 2; n1 is even and k = D(Cn1 ◦ Cn2 ) − 1; n1 is even and k = D(Cn1 ◦ Cn2 ); n1 is odd and k = D(Cn1 ◦ Cn2 ); n1 is odd and k = D(Cn1 ◦ Cn2 ) − 1.

Proof. By calculation. Theorem 2.1. Let Cn1 and Cn2 be the cycles with n1 > 4 and n2 > 4 vertices, respectively, then :  n3 2 2 2 1 if n1 even 8 (n2 + 1) + n1 (n2 + n2 ) − 2n1 n2 , W (Cn1 ◦ Cn2 ) = n31 n1 2 2 2 2 8

(n2 + 1) + n1 (n2 + n2 ) − 2n1 n2 −

8

(n2 + 1) , if n1 is odd

Proof. For n1 is even, we use the Lemma 2.1, and the theorem 1 in [10] : D(Cn1 ◦Cn2 )

W (Cn1 ◦ Cn2 )

= n(n − 1) − m +

X

(k − 2)dCn1 ◦Cn2 (k)

k=3

=

(n1 + n1 n2 )(n1 + n1 n2 − 1) + (D(Cn1 ◦ Cn2 ) − 3)(n1 n2 + n1 n22 )−

(n1 +2n1 n2 )+(D(Cn1 ◦Cn2 )−4)(

n1 n1 n22 +2n1 n2 +n1 n22 )+(D(Cn1 ◦Cn2 )−2) 2 2

D(Cn1 ◦Cn2 )−3

+(n1 + n1 n22 + 2n1 n2 )

X

(k − 2)

k=3

=

n31 (n2 + 1)2 + n21 (n22 + n2 ) − 2n1 n2 8

Similarly, when n1 is odd.

4223

Wiener index, hyper-Wiener index and degree distance index

2.2

The hyper-Wiener index of the Corona Cn1 ◦ Cn2

Theorem 2.2. Let Cn1 and Cn2 be the cycles with n1 > 4 and n2 > 4 vertices, respectively. Then : ( W W (Cn1 ◦Cn2 ) =

n41 48 (n2 n41 48 (n2

+ 1)2 + + 1)2 +

n31 2 16 (5n2 n31 2 16 (5n2

+ 6n2 + 1) + + 6n2 + 1) +

n21 2 24 (37n2 n21 2 48 (71n2

+ 26n2 + 1) − 72 n1 n2 , + 46n2 − 1) − n481 (15n22 + 186n2 + 3),

Proof. For n1 is even, we use the Lemma 2.1, and the theorem 2 in [10], then :

W W (Cn1 ◦ Cn2 )

D(Cn1 ◦Cn2 )

=

1 (3n(n − 1) − 4m + 2

=

1 ((3m + 3n1 n2 )(n1 + n1 n2 − 1) − 4(n1 + 2n1 n2 ) + ((D(Cn1 ◦ Cn2 ) − 2)2 + 2

X

(k 2 + k − 6)dCn1 ◦Cn2 (k))

k=3

D(Cn1 ◦Cn2 )

(D(Cn1 ◦Cn2 )−2)−6)(

n1 +2n1 n2 +n1 n22 )+ 2

(D(Cn1 ◦Cn2 )2 +D(Cn1 ◦Cn2 )−6)

X

n1 (k 2 +k−6)(n2 +1)2 +

k=3

mn2 +((D(Cn1 ◦Cn2 )−1)2 +(D(Cn1 ◦Cn2 )− 2

1) − 6)(n1 n2 + n1 n22 )) =

n31 n2 n4 7 (5n22 + 6n2 + 1) + 1 (37n22 + 26n2 + 1) + 1 (n2 + 1)2 − n1 n2 16 24 48 2

Similarly, when n1 is odd.

2.3

The degree distance index of the Corona Cn1 ◦ Cn2

Lemma 2.2. Let Cn1 and Cn2 be the cycles with n1 > 4 and n2 > 4 vertices, respectively, let u and v the vertices of Cn1 ◦ Cn2 which have as degree respectively n2 + 2 and 3, then : 

1 2 4 (n1 1 2 4 (n1

(

n21 4 (n2 n21 4 (n2

w(u, Cn1 ◦ Cn2 ) =

w(v, Cn1 ◦ Cn2 ) =

Proof. By calculation.

+ 4n1 n2 + n21 n2 ), if n1 even − 1)(n2 + 1) + n1 n2 , if n1 odd + 1) + n1 (2n2 + 1) − 4, + 1) + n1 (2n2 + 1) − 41 (n2 + 17),

if n1 even if n1 odd

if n1 even if n1 odd

4224

M. Essalih, M. El Marraki and A. Atmani

Theorem 2.3. Let Cn1 and Cn2 be the cycles with n1 > 4 and n2 > 4 vertices, respectively, then : ( DD(Cn1 ◦Cn2 ) =

n31 2 23 (2n2 n1 2 2 (2n2

+ 3n2 + 1) + n21 (7n22 + 5n2 ) − 12n1 n2 , + 3n2 + 1) + n21 (7n22 + 5n2 ) − n21 (2n22 + 27n2 + 1),

if n1 even if n1 odd

Proof. For n1 is even, we use the Lemma 2.1, the Lemma 2.2, and the theorem 3 in [10], then : DD(Cn1 ◦ Cn2 ) =

X

w(u, Cn1 ◦ Cn2 )deg(u)

u∈G

X

=

u∈Cn1 ◦Cn2

w(u, Cn1 ◦ Cn2 )deg(u) +

X

w(v, Cn1 ◦ Cn2 )deg(v)

v∈G

= mw(u, Cn1 ◦ Cn2 )(n2 + 2) + 3n1 n2 w(v, Cn1 ◦ Cn2 ) =

n31 (2n22 + 3n2 + 1) + n21 (7n22 + 5n2 ) − 12n1 n2 2

Similarly, when n1 is odd.

3

Conclusion

Finally it might be observed that the corona operation allows us to obtain another graph whose group is in general isomorphic to the wreath product of the groups of the graphs G1 and G2 , a sufficient condition being this time only that G2 have no isolated points. A unified approach to the Wiener topological index and its various recent modifications, is presented. Among these modifications particular attention is paid to the Kirchhoff, Harary, Szeged, Cluj and Schultz indices, as well as their numerous variants and generalizations. Relations between these indices are established and methods for their computation are described. The mutual correlation of the proposed topological indices and the correlation of these indices with physico-chemical properties of molecules, are examined. In conclusion, in this paper we have presented the Corona of two graphs Cn1 and Cn2 , and some topological indices, like the index Wiener W (Cn1 ◦ Cn2 ), the degree distance index DD(Cn1 ◦ Cn2 ) and the hyper-Wiener index W W (Cn1 ◦ Cn2 )) of it.

Wiener index, hyper-Wiener index and degree distance index

4225

References [1] S. Alikhani and Y. Peng, Introduction to Domination Polynomial of a Graph, arXiv (Math.CO), 1 (2009), 1-10. [2] C. Barrientos, Craceful labelings of chain and corona graphs, Bulletin of the ICA, 34 (2002), 17-26. [3] H. Cheng, J. Yang, Hosoya index of the Corona of Two Graphs, South Asian Journal of Mathematics, 2 no. 2 (2012), 144-147. [4] M.V. Diudeaa and I. Gutman, Wiener-Type Topological Indices, CROATICA CHEMICA ACTA, 71 no.1 (1998), 21-51. [5] M. V. Diudea, I. Gutman and L. Jantschi, Molecular Topology , first Eddition, Cluj and Kragujevac, Fall 1999. [6] M. Essalih, M. El Marraki and G. Al hagri,Some topological indices of Spider’s Web Planar Graph, Applied Mathematical Sciences (AMS), 6 no. 63 (2012), 3145-3155. [7] M. Essalih, M. El Marraki and G. Al hagri, Calculation of some topological indices, Journal of Theoretical and Applied Information Technology (JATIT), 30 no.2 (2011), 122-127. [8] R. Fruchet and F. Harary,On the Corona of Two Graphs, AEQ. Math. 4 (1970), 322-325. [9] C.E. Go, Domination in the Corona and Join of Graphs, International Mathematical Forum 6 no.16 (2011), 763-771. [10] G. Hagri, M. El Marraki and M. Essalih,The Degree Distance of Certain Particular Graphs, Applied Mathematical Sciences (AMS), 6 no. 18 (2012), 857-867. [11] H. Hosoya,Topological Index and some counting polynomials for characterizing the topological structure and properties of Molecular graphs, Research of pattern Formation, edited by R. Takaki, 63-75. [12] V. KBalakrishnan, Graph theory, Schaum’s outline, Mcgraw-hill companies, 1995.

4226

M. Essalih, M. El Marraki and A. Atmani

[13] B. Lin, H. Cheng, J. Yang and F. Xia, The Wiener index of the corona two graphs Cm ◦ Cn , South Asian Journal of Mathematicas, 2 no.2 (2012), 122-125. [14] G. Sabidussi, The composition of graphs, Duke Math. J, 26 (1959), 693696. [15] H. Wang and G. Yu, All but 49 numbers are Wiener indices of trees, Acta Appl.Math, 92 (2006), 15-20. [16] D.B. West, Introduction to Graph Theory , Second Eddition, Prentice Hall (2002). [17] S. WU, J. YANG and H. CHENG Merrifield-simmons index of the corona of two graphs, South Asian Journal of Mathematics, 2 no. 3 (2012), 274278. Received: March 20, 2014