Mathematics and Computer Science
Volume 1, Issue 4, November 2016, Pages: 101-107

Computing Energy and Some Topological Indices of

1Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

2Young Researchers and Elite Club, Karj Branch, Islamic Azad University, Karaj, Iran

(M. J. Nikmehr)
(N. Soleimani)

*Corresponding author

Mohammad Javad Nikmehr, Najmeh Soleimani. Computing Energy and Some Topological Indices of . Mathematics and Computer Science. Vol. 1, No. 4, 2016, pp. 101-107. doi: 10.11648/j.mcs.20160104.15

Received: September 3, 2016; Accepted: November 16, 2016; Published: December 20, 2016

Abstract: For a commutative ring , the total graph of  which denoted by , is a graph with all elements of R as vertices, and two distinct vertices  are adjacent if and only if , where  denotes the set of zero-divisors of R. In an earlier study, we computed Wiener, hyper-Wiener, reverse Wiener, Randi, Zagreb,  and  indices of zero-divisor graph. In this study, some computer programs are prepared to calculate the zero-divisors and adjacency matrix of the given graph which, apply these programs to compute the energy and first edge-Wiener, sum-connectivity, harmonic, augmented Zagreb and hyper-Zagreb indices.

Keywords: Ring, Zero-Divisor Graph, MATLAB Program, Energy, Topological Indices

Contents

1. Introduction

A graph  consists of a set of vertices  and a set of edges . The number of vertices and edges in a graph will be denoted by  and , respectively. The graph  is connected if each pair of the vertices in  belongs to a path. The degree of the vertex, written as , is the number of edges with  as an end vertex. For two vertices  we define their distance  as the length of any shortest path connecting  and  in . A regular graph is a graph where each vertex has the same number of neighbors. A regular graph with vertices of degree  is called a -regular graph or regular graph of degree . Let  be a commutative ring with identity. The total graph of  denoted by , is a graph with the vertex set  such that two vertices  and  are adjacent if and only if . We denote the set of zero-divisors and unit elements of  by  and , respectively. This concept was firstly introduced in [1] by Anderson and Badawi. Also, the zero-divisor graph of , denoted as . Throughout this study, let  that  be a positive integer, where for , ,  is a distinct prime number and  and suppose that  for some positive integer . Mathematical calculations are absolutely necessary to explore important concepts in chemistry. In chemical graph theory, there are many molecular descriptors (or topological indices) for a connected graph, that have very useful properties to study of chemical molecules. The use of topological indices as structural descriptors is important in proper and optimal nanostructure design. In a series of studies some topological indices are computed for nanostructures and some graphs [2-7]. This article is the continuation of the work [8], which were provided MATLAB programs, to calculate the energy and first edge-Wiener, sum-connectivity, harmonic, augmented Zagreb and hyper-Zagreb index of the given zero-divisor graph.

2. Methods

In this section, we recall some algebraic definitions related to the topological indices chosen for the present study. Chemical graph theory interdisciplinary science that applies graph theory to the study of molecular structures. The molecules or chemical compounds are modeled by an undirected graph named molecular graphs.

Equivalence between chemical and graph theoretical terms:

Atom Vertex

Molecule Molecular graph

Covalent bond Edge

Valency of an atom  Vertex degree

Skeletal structure  Hydrogen-depleted graph

Energy level  Eigenvalue

Example 2.1 In order to make it easier for readers to understand the molecular graph, an example is given on a hydrocarbon, together with its hydrogen-depressed graph as shown in Figure 1.

Figure 1. Structural formula and molecular graph of benzene, .

The adjacency matrix of a molecular graph  with  vertices is an  matrix  defined by: , if vertices  and are connected by an edge and, , otherwise. The energy of a graph , often denoted , is defined to be the sum of the absolute value of the eigenvalues of its adjacency matrix. Hence if  is the adjacency matrix of , and  are the eigenvalues of A(G), then . In chemistry the energy of a graph is extensively studied since it can be used to approximate the total -electron energy of the molecule represented by that graph, for more details see [9, 10]. A topological index is a numerical parameter mathematically derived from the graph structure. The topological indices of molecular graphs are widely used for establishing correlations between the structure of a molecular compound and its physic-chemical properties or biological activity (e.g., pharmacology).

The Wiener index is the oldest and one of the most studied distance-based topological indices in chemistry. It was introduced by Harold Wiener [11], equal to the sum of distances between all pairs of vertices of the graph under consideration. Other much studied distance-based topological indices are edge Wiener indices. Let. Set  and

The first edge-Wiener index ( was introduced by Iranmanesh et al. [12] as follow:

(1)

Also, vertex-degree-based topological indices discussed in this study:

Sum-connectivity (X), [13]:    (2)

Harmonic (H), [14]:          (3)

Augmented Zagreb (AZI), [15]:    (4)

Hyper-Zagreb (HM), [16]:     (5)

Example 2.2 Let  be a defect carbon nanocone. To obtain topological indices, starting from molecular structure, viewed as a graph, and considering the number of the vertices and edges. Obviously, for  (see Figure )  and . Now, it is easy to see that  has  vertices and  edges. We can see that the edge set of graph can be dividing to three partitions edges with endpoints , edges with endpoints  and edges with endpoints . By use an algebraic method we obtain ,  and .

Figure 2. The graph of defect carbon nanocone,  with  and .

Now, we calculate the Sum-connectivity index, Harmonic index, Augmented Zagreb index and hyper-Zagreb index for the molecular graph of Figure 2 by use an algebraic method:

a. .

b. .

c. .

d. .

3. Results and Discussion

The goal of this section is to compute the topological indices of the zero-divisor graph . Begin with several properties of the zero-divisor graph, which will be helpful to express of main results.

Theorem 3.1 [17] Let  and  be a natural number. If , then the followings hold:

a. If  is prime, then , ().

b. If  is odd and not a prime number, then

c. If  is even, then  and  is -regular.

The following result is a consequence of Theorem 3.1.

Corollary 3.1 For a positive integer , we have:

Note 3.1 For some values of the  is not connected. Indeed,  is not connected if and only if , where  is prime and . The following figures give examples of , for some values of .

Example 3.1 The following figures represent some zero-divisor graphs in which Figure 3 and 4 are connected.

Figure 3. .

Figure 4. .

Example 3.2 The following figures represent some zero-divisor graphs in which Figure 5 and 6 are not connected.

Figure 5. .

Figure 6. .

First MATLAB program is introducing a particular commutative ring  and we will discuss the value  for . Since for the calculation of topological indices the graph should be connected, therefore ,  is prime and .

Program 3.1 A MATLAB program for checking different values  of :

function [flag, Primes, Powers, N]=CHECK()

N=input('Input an integer: ');

IsPrime=isprime(1: N);

Primes=find(IsPrime);

Powers=zeros(size(Primes));

M=N;

for k=1: numel(Primes)

while mod(M, Primes(k))==0

Powers(k)=Powers(k)+1;

M=M/Primes(k);

end

end

if (sum(Powers~=0)>1)

flag=true;

else

flag=false;

end

Program 3.2 A MATLAB program compute the set zero divisors for , which is very useful in the other programs:

function [m]=ZeroDivisor(N, Powers, Primes)

m=[];

m(1)=0;

if((numel(Primes)>1)&&(sum(Powers~=0)>1))

for i=1: numel(Primes)

n=1;

if(Powers(i)>0)

while Primes(i)*n<N

if(m~=Primes(i)*n)

m=[m, Primes(i)*n];

end

n=n+1;

end

end

end

end

m=sort(m);

end

Theorem 3.2 [18] Let  be an arbitrary graph. Then

Remark 3.1 By a well-known result on the subject of graph energy,

,

for -vertex -regular graph .

Theorem 3.3 Let  be a zero-divisor graph. Then the followings hold:

a. If  is prime, then .

b. If  is odd and not a prime number, then

c. If n is even, then

Proof. By Corollary 3.1, Theorem 3.2 and Remark 3.1, we are done.

Program 3.3 A MATLAB program for finding adjacency matrix of :

A=zeros(N);

for i=1: N

for j=i+1: N

if sum(m==i-1)==1

if sum(m==j-1)==1

jam=i+j-2;

if (sum(m==jam )==1|| jam==N || sum(m==jam-N)==1)

A(i, j)=1;

end

else

jam=i+j-2;

if (sum(m==jam )==1|| jam==N || sum(m==jam-N)==1)

A(i, j)=1;

end

end

else

jam=i+j-2;

if (sum(m==jam )==1|| jam==N || sum(m==jam-N)==1)

A(i, j)=1;

end

end

end

end

for r=N:-1:1

for i=N-1:-1:1

A(r, i)=A(i, r);

end

end

end

We apply Program 3.3 to compute the energy, first edge-Wiener index and vertex-degree-based topological indices of the zero-divisor graph .

Program 3.4 A MATLAB program for computing the energy of :

clc;

clear;

[flag, Primes, Powers, N]=CHECK();

if(flag)

[m]=ZeroDivisor(N, Powers, Primes);

disp(['The number of zero divisors=', num2str(numel(m))]);

e=eig(A)

energy=sum(abs(e));

disp(['Energy=', num2str(sum(abs(e)))]);

else

disp('wrong input');

end

Using this program, we have:

Table 1. The Values of  for .

Program 3.5 A MATLAB program for finding distance matrix of :

function D=dismat(m, N)

D=zeros(N);

m=sort(m);

for i=1: N

for j=i+1: N

if sum(m==i-1)==1

if sum(m==j-1)==1

jam=i+j-2;

if (sum(m==jam )==1|| jam==N || sum(m==jam-N)==1)

D(i, j)=1;

else

D(i, j)=2;

end

else

jam=i+j-2;

if (sum(m==jam )==1|| jam==N || sum(m==jam-N)==1)

D(i, j)=1;

else

D(i, j)=2;

end

end

else

jam=i+j-2;

if (sum(m==jam )==1|| jam==N || sum(m==jam-N)==1)

D(i, j)=1;

else

D(i, j)=2;

end

end

end

end

for r=N:-1:1

for i=N-1:-1:1

D(r, i)=D(i, r);

end

end

end

Program 3.6 A MATLAB program for finding minimum of a vector of numbers:

function [k]=mini(V)

[m, n]=size(V);

k=V(1);

for i=2: n

if V(i)<k

k=V(i);

end

end

end

Program 3.7 The program compute the first edge-Wiener index of . This program includes several sub programs:

clc;

clear;

[flag, Primes, Powers, N]=CHECK();

c=1;

for i=1: numel(Primes)

if Powers(i)~=0

primes2(c)=Primes(i);

c=c+1;

end

end

if(flag)

[m]=ZeroDivisor(N, Powers, Primes);

D=dismat(m, N);

k=0; We=0; We0=0;

for i=1: size(A, 1)

for j=1: size(A, 1)

if i < j & A(i, j)==1

k=k+1;

x(k)=i;

y(k)=j;

end

end

end

for i=1: k

for j=1: k

if i~=j

F=[D(x(i), x(j)), D(x(i), y(j)), D(y(i), x(j)), D(y(i), y(j))];

We(i, j)=mini(F)+1;

else

We(i, j)=0;

end

end

end

for i=1: k

for j=1: k

We0=We0+(We(i, j)/2);

end

end

disp(['first edge-Wiener index=', num2str(We0)]);

else

disp('wrong input');

end

Lemma 3.1 Let  be an arbitrary graph. Then  is -regular if and only if one of the followings holds:

a. .

b. .

c. .

d. .

Proof. The proof is clear.

In order to provide a unified approach to the results discussed in next theorem, we express the following notation.

Note 3.2 Let  be an arbitrary edge. Three cases may occur:

(i)

Since the degree of a vertex  is , then the number of is .

(ii)  and .

Since the degree of a vertex  is , the number of unit vertices adjacent to a vertex  is . So, the coefficient of  is

(iii) .

Similar to the previous cases, we have

Now, we calculate the sum-connectivity index, harmonic index, augmented Zagreb index and hyper-Zagreb index of  by use an algebraic method.

Theorem 3.4 It holds that:

a.

b.

c.

d.

Proof. The formulas follow immediately from apply the precedent definitions, Lemma 3.1 and Note 3.2.

Program 3.8 A MATLAB program for finding degree of a vertex:

function [z]=degv(A, m)

z=0;

for a=A(m,:)

z=z+a;

end

end

Program 3.9 We offer a MATLAB program for calculating vertex-degree-based topological indices of :

clc;

clear;

[flag, Primes, Powers, N]=CHECK();

c=1;

for i=1: numel(Primes)

if Powers(i)~=0

primes2(c)=Primes(i);

c=c+1;

end

end

if(flag)

[m]=ZeroDivisor(N, Powers, Primes);

k=0; X=0; H=0; AZI=0; HM=0;

for i=1: size(A, 1)

for j=1: size(A, 1)

if i < j & A(i, j)==1

k=k+1;

x(k)=i;

y(k)=j;

end

end

end

for i=1: k

X=X+(1/sqrt((degv(A, x(i))+degv(A, y(i)))));

H=H+(2/(degv(A, x(i))+degv(A, y(i))));

AZI=AZI+((degv(A, x(i))*degv(A, y(i)))/(degv(A, x(i))+degv(A, y(i))-2))^3;

HM=HM+((degv(A, x(i))+degv(A, y(i)))^2);

end

disp(['Sum-connectivity index=', num2str(X)]);

disp(['Harmonic index=', num2str(H)]);

disp(['Augmented Zagreb index=', num2str(AZI)]);

disp(['Hyper-Zagreb index=', num2str(HM)]);

else

disp('wrong input');

end

To investigate the efficiency of Program 3.7 and Program 3.9, we consider two graphs which depicted in Figure 3 and 4.

Example 3.3 The values of  and  for  are as follows:

Table 2. Some values of the topological indices of .

4. Conclusion

In fact, we made a connection between commutative ring theory, graph theory and topological indices. The main purpose of this study is to extend some results given in [8]. We have mentioned here some MATLAB programs, to calculate the adjacency matrix of the given graphs. Then we compute the energy and some topological indices by these programs.

Acknowledgments

The authors would like to thank the anonymous referee for his/her helpful comments that have improved the presentation of results in this article.

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