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International Journal of Advanced Research in Engineering and Technology

(IJARET) Volume 7, Issue 3, May–June 2016, pp. 56–65, Article ID: IJARET_07_03_005

Available online at

http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=7&IType=3

ISSN Print: 0976-6480 and ISSN Online: 0976-6499

© IAEME Publication

INDEPENDENT DOMINATION NUMBER OF

EULER TOTIENT CAYLEY GRAPHS AND

ARITHMETIC GRAPHS

S.Uma Maheswari

Lecturer in Mathematics, J.M.J.College,

Tenali, Andhra Pradesh, India

B.Maheswari

Department of Applied Mathematics, S.P. Women’s University,

Tirupati, Andhra Pradesh, India

ABSTRACT

Nathanson was the pioneer in introducing the concepts of Number Theory,

particularly, the “Theory of Congruences” in Graph Theory, thus paved the

way for the emergence of a new class of graphs, namely “Arithmetic Graphs”.

Cayley graphs are another class of graphs associated with the elements of a

group. If this group is associated with some arithmetic function then the

Cayley graph becomes an Arithmetic graph.

In this paper, we study independent domination number of Euler totient

Cayley graphs and Arithmetic graphs.

Key words: Dominating set, Independent dominating set, Euler totient Cayley

graph, Arithmetic graph.

AMS subject classification: 05C69

Cite this article: Uma Maheswari S. and Maheswari B. Independent

Domination Number of Euler Totient Cayley Graphs and Arithmetic Graphs.

International Journal of Advanced Research in Engineering and Technology,

7(3), 2016, pp 56–65. http://www.iaeme.com/IJARET/issues.asp?JType=IJARET&VType=7&IType=3

1. INTRODUCTION

The theory of domination was formalized by Berge [3] and Ore [9] in 1962. Since

then it has developed rapidly and various variations of domination are introduced and

studied. The independent domination number and the notation were introduced by Cockayne and Hedetniemi in [4, 5] and later developed by Allan and Laskar [1].

Independent dominating sets have been studied extensively in the literature [2, 6, 7]

Independent Domination Number of Euler Totient Cayley Graphs and Arithmetic Graphs


A dominating set of a graph is a subset of vertex set of such that every

vertex in is adjacent to at least one vertex in . The minimum cardinality of a

dominating set of is called the domination number of and is denoted by

A subset of vertices of of a graph is called an independent set if no two

vertices in it are adjacent. An independent dominating set of is a set that is both

dominating and independent in . The independent domination number of , denoted

by , is the minimum cardinality of an independent dominating set.

2. EULER TOTIENT CAYLEY GRAPH AND ITS

PROPERTIES

The concept of Euler totient Cayley graph is introduced by Madhavi [8] and studied

some of its properties. For any positive integer , let be the

residue classes modulo . Then , where addition modulo is is an abelian

group of order

The number of positive integers less than and relatively prime to is denoted by

and is called an Euler totient function. Let denote the set of all positive

integers less than and relatively prime to that is

Then

The Euler totient Cayley graph is defined as follows.

The Euler totient Cayley graph is defined as the graph whose vertex set

V is given by and the edge set is

Clearly as proved by Madhavi [8], the Euler totient Cayley graph is

a connected, simple and undirected graph,

( ) - regular and has

edges,

Hamiltonian,

Eulerian for

bipartite if is even and

Complete graph if is a prime.

3. ARITHMETIC GRAPH

The concept of Arithmetic graph is introduced by Vasumathi [10] and studied some of its properties.

Let be a positive integer such that

. Then the Arithmetic

graph is defined as the graph whose vertex set consists of the divisors of and two

vertices are adjacent in graph if and only if GCD for some prime

divisor of

In this graph the vertex 1 becomes an isolated vertex. Hence we consider the

Arithmetic graph without vertex 1 as the contribution of this isolated vertex is nothing when the properties of these graphs and enumeration of some domination

parameters are studied.

Clearly, graph is a connected graph. Because if is a prime, then graph consists of a single vertex. Hence it is a connected graph. In other cases, by the

definition of adjacency in there exist edges between prime number vertices, their

Uma Maheswari S. and Maheswari B.


prime power vertices and also their prime product vertices. Therefore each vertex of

is connected to some vertex in this graph is denoted by

4. INDEPENDENT DOMINATING SETS OF EULER TOTIENT

CAYLEY GRAPH We determine minimum independent dominating sets and independent domination

number of graph as follows.

4.1. Theorem

If is a prime, then the independent domination number of is 1.

4.1.1 Proof

Let be a prime. Then is a complete graph.

Let where is any vertex in V. Then every is adjacent to vertex Thus every vertex in is adjacent to so that forms a dominating set in

since it is evident that is a minimum dominating set in

In fact every singleton vertex set forms a minimum dominating set and also

becomes an independent dominating set of .

Thus

4.2. Theorem

If

then the independent domination number of

is

.

4.2.1. Proof

Suppose

, where , are distinct primes and are

integers ≥ 1. Consider the following sets in

For , we shall show that each of the above sets, say

is an independent set of .

Let , Then and where .

Now and GCD

since . So Hence and are not adjacent. This shows that no two

vertices in are adjacent. So becomes an independent set of



By the construction of the sets , it is obvious that for 1

and This shows that the vertex set is the union of disjoint subsets

which are independent and

.

By the construction of the sets it is obvious that each is a maximal

independent set of but every maximal independent set is a minimal

dominating set. So, each of the sets is an independent dominating set with

minimum cardinality. Hence

5. INDEPENDENT DOMINATING SETS OF ARITHMETIC

GRAPH

We determine minimum independent dominating sets and independent domination

number of graph as follows.

5.1. Theorem

If

, where , , are primes and are integers ≥ 1, then

the independent domination number of is given by

Where is the core of .

5.1.1. Proof

Suppose

Consider the graph with vertex set we have the

following cases.

5.1.2. Case 1

Suppose for all . That is

where then we show that

the set becomes an independent dominating set of .

By the definition of graph, it is obvious that the vertices in are

primes , their powers and their products.

All the vertices , for which GCD are adjacent to the

vertex in All the vertices , for which GCD are

adjacent to the vertex in Continuing in this way we obtain that all the vertices

, for which GCD are adjacent to the vertex in Since

every vertex in has atleast one prime factor viz., ( as they

are divisors of every vertex in is adjacent to at least one vertex in Thus

becomes a dominating set of .

We now prove that is minimum. Suppose we remove any from then the

vertices of the form , will be non-adjacent to any other vertex as

GCD for Therefore every , must be included

into If we form a minimum dominating set in any other manner, the order of such a

set is not smaller than that of This follows from the properties of prime divisors of

a number.Thus becomes a minimum dominating set of .

Now we show that is an independent set. Consider any two vertices , in

or , these vertices are not adjacent to each other because GCD .



Hence becomes an independent dominating set of with minimum cardinality

h

Hence

5.1.3. Case 2

Suppose for only one That is, is the only prime divisor of with exponent

1. Then

Then as in Case 1 we can see that is a minimum dominating set

of which is also independent.

Hence

5.1.4. Case 3

Suppose for more than one Denote the prime divisors of with exponent 1

by and write these primes in ascending order. Then we have

Let

Then we show that forms a minimum dominating set of Any vertex in

will be of the form

where and

for Then clearly is a dominating set as every vertex in

is adjacent to at least one vertex in However this is not a minimum

dominating set.

Let where the vertices are adjacent to the

vertex . This is clearly a dominating set of and deletion of vertices in this set will not make it a dominating set any more.

By properties of prime numbers no two vertices in the set are adjacent. Hence

becomes an independent dominating set of with minimum cardinality.

Hence .

6. ILLUSTRATIONS



Euler Totient Cayley Graph

Minimum Independent

Dominating Set {0} {0,7} {0,5,10,15,20} {0,5,10,15,20,25}

1 2 5 6



Arithmetic Graph

=

Minimum

Independent

Dominating Set

{2,15} {2,5} {2,3,5} {2,3,35}

2 2 3 3

ACKNOWLEDGEMENT

This work is a part of Minor Research Project of University Grants Commission with

Ref. No.F MRP-5510 /15 (SERO/UGC).

REFERENCES

[1] Allan,R.B. and Laskar, R.-On domination and independent domination numbers

of a graph.Discrete Math., 23:73–76, 1978.

[2] Ao, S., Cockayne, E.J., Mac Gillivray, G. and Mynhardt, C.M.- Domination

critical graphs with higher independent domination numbers. J. Graph Theory,

22:9–14, 1996.

[3] Berge, C. - Theory of Graphs and its Applications. Methuen, London, 1962.

[4] Cockayne, E.J. and Hedetniemi, S.T.- Independence graphs. Congr. Numer.

X:471–491,1974.

[5] Cockayne, E.J. and Hedetniemi, S.T.- Towards a theory of domination in

graphs.Networks, 7:247–261, 1977.

[6] [6]Duckworth, W. and Wormald, N.C.- On the independent domination number

of random regular graphs. Combin. Probab. Comput., 15:513–522, 2006.



[7] Lam, P.C.B., Shiu, W.C. and Sun, L.-On independent domination number of

regular graphs. Discrete Math., 202:135–144, 1999.

[8] Madhavi, L.-Studies on domination parameters and enumeration of cycles in

some Arithmetic Graphs, Ph.D. Thesis submitted to S.V. University, Tirupati,

India, 2002.

[9] Ore, O. - Theory of graphs. Amer. Math. Soc. Transl., 38:206–212, 1962.

[10] Vasumathi, N. - Number theoretic graphs, Ph.D. Thesis submitted to S.V.

University, Tirupati, India, 1994.

independent domination number of euler ...cayley graph becomes an arithmetic graph. in this paper,...

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