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 wi th the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an Arithmetic graph.

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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]

2. Independent Domination Number of Euler Totient Cayley Graphs and Arithmetic Graphs
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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

3. Uma Maheswari S. and Maheswari B.
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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

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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 .

5. Uma Maheswari S. and Maheswari B.
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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

6. Independent Domination Number of Euler Totient Cayley Graphs and Arithmetic Graphs
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7. Uma Maheswari S. and Maheswari B.
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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

8. Independent Domination Number of Euler Totient Cayley Graphs and Arithmetic Graphs
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9. Uma Maheswari S. and Maheswari B.
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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.

10. Independent Domination Number of Euler Totient Cayley Graphs and Arithmetic Graphs
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[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.