Edge Cover Domination in Mangoldt Graph

In their recent study of arithmetic graphs associated with certain arithmetic functions, the authors have introduced a new class of arithmetic graphs associated with Mangoldt function Λ(n), n ≥ 1, an integer and studied their basic properties and vertex cover. In this paper the edge cover, edge domination set, bondage number, non – bondage number and their parameters have been obtained for these graphs.


INTRODUCTION
It was during 1850's, a study of dominating sets in graphs started purely as a problem in the game of chess.Chess enthusiasts in Europe considered the problem of determing the minimum number of queens that can be placed on a chess board so that all the squares are either attacked by a queen or occupied by a queen.A precise notion of a dominating set, that is present in the current literature can be said to be given by Berge (1962) and Ore (1962) Since then a number of graph theorists, Allan and Laskar (1978) Allan et al., (1984); Cockayne and Hedetniemi (1977); Haynes and Slater (1998); Kulli and Sigarkant (1992) etc., have studied various domination parameters of graphs.In a graph G the entire vertex set V trivially covers every edge of G in the sense that every edge of G is incident with some vertex of V.In practical problems one may be interested in a subset of V with minimum cordinality which covers every edge of G.In the same way a subset of the edge E of G with minimum cordinality which covers every vertex of G plays a crucial roll.These considerations lead to the concepts of vertex cover, vertex covering number, edge cover, edge covering number and other domination parameters of a graph, which constitute the domination theory of graphs.Domination theory of graphs has many applications in Engineering and Communication Networks.For these applications, the arithmetic graphs associated with certain number theoretic arithmetic functions like the Euler totient function Φ(n), the divison function d(n), the quadratic residue function and the Mangoldt function Λ(n), n ≥ 1, an integer come handy.These arithmetic graphs have been studied in detail by Madhavi (2003).Maheswari and Madhavi (2008) have studied the basic properties and vertex cover of the Mangoldt graph associated with Mangoldt function Λ(n).This paper is devoted for the study of edge domination of a Mangoldt graph.In this study, we have followed Bondy and Murty (1979);and Harary (1969) for graph theory and Apostol (1989) for number theory terminology and notations not explained here.

THE MANGOLDT GRAPH AND ITS PROPERTIES
In this section we introduce a new class of arithmetic graph, namely, Mangoldt graph associated with the Mangoldt arithmetic function Λ(n), n ≥ 1 an integer (Apostol, 1989) and briefly outline its basic properties, whose proofs can be found in Maheswari and Madhavi (2008).

Definition
Let n ≥ 1 be an integer.The Mangoldt function Λ (n) is defined as follows: log p if n = p m for some prime p and some m ≥ 1, 0 otherwise.

Definition
Let n ≥ 1 be an integer.The Mangoldt graph M n is defined as the graph whose vertices are the elements of the set {1, 2, …., n } and two distinct vertices x, y are adjacent (or( x, y) is an edge) if and only if Λ ( x .y ) = 0 or x .y is not a power of a prime.
The following theorem gives a formula for the number of edges in M n which is useful in finding the non -bondage number of M n in section 4.

Theorem
Let n ≥ 2 be an integer, In Maheswari and Madhavi (2008) the authors have obtained the basic properties of the Mangoldt graph and studied its vertex domination parameters.For completeness they are stated below without proofs, which can be found in Maheswari and Madhavi (2008).

Theorem
For n ≤ 5, M 5 is a disconnected graph.

Theorem
For n ≥ 6, M n is neither a bipartite graph nor a tree.

Theorem
For n ≥ 6, the graph M n is a connected graph.

Theorem
For n ≥ 10, M n is Hamiltonian.

Definition
A set S of vertices of a graph G is called a vertex cover of G if every edge of G is incident with some vertex in S. A minimum vertex cover is the one with minimum cardinality and the cardinality of a minimum vertex cover of a graph G is called the covering number of G and it is denoted by β (G).

Definition
A dominating set of graph G is a subset D of the vertex set V such that each vertex of V-D is adjacent to at least one vertex of D. A minimum dominating set is one with minimum cardinality.The domination number γ(G) is the cardinality of a minimum dominating set.

Theorem
If n ≥ 6, then for each positive integer r ≤ n which is not a power of a single prime, the singleton set {r} is a minimum dominating set of M n and γ (M n ) = 1.

EDGE DOMINATION IN MANGOLDT GRAPH
Just as a (minimum) vertex cover in a graph is the set of vertices in G which (minimum cardinality and) covers every edge of G, a (minimum) edge cover of a graph is the set of edges (with minimum cardinality and ) covers all vertices of the graph.

Definition
A set F of edges of a graph G is called an edge cover of G if every vertex of G is incident with some edge in F. A minimum edge cover is one with minimum cardinality.The number of edges in a minimum edge cover of G is called the edge covering number of G and it is denoted by β | (G).
If n is an even number which is not a power of 2 then the set of edges F = {(2,3),(4,5), . .., (n-2, (n-1), (n,1)}forms an edge covering of M n .As above it can be seen that F is a minimum edge cover of M n with cardinality n/2.
If n is an odd number then for any composite number r ≤ n which is not a power of a prime, the set of edges F={(2,3), (4,5),…, (n-1,n), (r,1)} is an edge cover of M n with cardinality (½)(n-1)+1 = (n+1)/2.On the other hand since n is odd, if n vertices of M n are paired into (n-1)/2 distinct pairs of distinct vertices, one vertex is leftout.So the minimum number of edges one needs to cover all the vertices of M n is ( ½) (n-1)+1 = (n+1)/2 = |F| so that F is a minimum edge cover of M n .■ The following corollary is immediate.

Corollary
If n ≥ 6, the edge covering number is n/2 if n = 2 r , r > 1 or n is even and n ≠ 2 r and (n+1)/2 if n is odd.
Another concept which plays a crucial role in the applications of graph theory is the edge dominating set.This is the set of edges of a graph G such that every edge of the graph not in this set is adjacent to atleast one edge of this set.

Definition
A subset F of the edge set E in a graph G is an edge dominating set of G if each edge of E not in is adjacent to at least one edge in F. The minimum cardinality among all edge dominating sets of G is called an edge domination number of G and is denoted by γ | (G) .
Let (r, s)∈E-E 1 .If r = 1 then s must be a product of powers of at least two distinct primes.
If s is even then (s−1, s)∈ E 1 and it is adjacent to (r, s).
If s is odd then (s, s+1)∈ E 1 and it is adjacent to (r, s).
If r = 2 then s must be either a prime ≠ 2 or a product of distinct primes.
In the first case if s > 2 is a prime then the edge (s, s+1) is in E 1 and it is adjacent to (r,s).In the second case if s is a product of distinct primes then (r,s) is adjacent to (s, s+1) in E 1 when s is even or adjacent to (s, s+1) in E 1 when s is odd.So let r ≥ 3 and s ≠ r+1.Suppose r is odd.Then r =2t+1 for some positive integer t and the edge (2t+1, 2t+2) in E 1 is adjacent with (r, s).
Suppose r is even.Then r =2t for some integer t ≥ 1 and the edge (2t, 2t−1) in E 1 is adjacent with (2t, s).In this case E 1 is an edge dominating set of M n .Any edge in E 1 is not adjacent to any one of the remaining edges in E 1 since, if (i,i+1), (j, j+1), i ≠ j are any two edges in E 1 then i≠ j, j+1 and j ≠ i, i+1.So, if we delete the edge (i, i+1) from E 1 then the remaining edge set is the minimum edge dominating set.
The following corollary is immediate.

Remarks
In

THE BONDAGE NUMBER OF A MANGOLDT GRAPH
Deletion of a set F of edges from a graph G (retaining the end vertices) may result in increasing the vertex covering number of G.Among all these sets F, sets with minimum cardinality have importance and this leads to the concept of bondage number of G.

Definition
Let F be a set of edges of G such that γ(G−F) > γ(G).Then the bondage number b (G) of G is defined to be the minimum number of edges in F.

Graph of M 10 -F
The number theoretic lemma given below gives a positive integer N associated with a given integer n ≥ 2. The bondage number of M n is given in terms of N.

Theorem
For n ≥ 6 the bondage number b(M n ) is ⎡N/2⎤ where ⎡N/2⎤ is the smallest integer > N/2 and N is the positive integer given in Lemma 4.3.
Proof: Let {m 1 , m 2, m 3 , ..…, m N } be the set of all integers ≤ n which are products of two or more distinct prime powers and let It is also a minimum dominating set.For, if we delete from is easy to see that either the singleton set { } or {m i m i + 1 } is a dominating set of (M n -F 1 ) and thus or the domination number is not increased.Thus F is a minimum set of edges of M n such that γ (M n -F) > γ (M n ) so that the bondage number b(M n ) =⎪F⎪= ⎡N/2⎤.
The singleton set {6} is not a vertex cover of M 15 -F, since 10 and 15 are not adjacent to 6 in this graph.In the same way each of the other singleton sets are not vertex covers of M 15 -F.

Definition
The non -bondage number b ˈ (G) of a graph G is the maximum cardinality among all sets of edges F⊆ E such that γ (G -F) = γ(G).

Theorem
For n ≥ 6, the non -bondage number b \ (M n ) of the Mangoldt graph M n is ε -(n-1) where ε is the number of edges of M n .
Proof: Let V and E respectively denote the vertex set and edge set of M n so that M n = G (V,E).
Suppose E 1 is the maximum set of edges in G such that γ (G (V, E-E 1 )) = γ(G).
For any composite number r which is a product of more than two distinct prime powers, the set r,s)|1≤s ≤ n, s ≠ r} of edges is such that γ (G (V, E 2 )) = γ(G).Since | E 2 |= n-1 we have The following example gives the importance of the theorem 4.10 in determining the nonbondage number of M n for any integer n ≥ 6.

Example
Consider the graph M 15 given in the Example 4.5.By the Theorem 2.3 the number of edges in

CONCLUSION
In Maheswari and Madhavi (2008) it is shown that the Mangoldt graph M n is Hamiltonian for n ≥ 10.Maheswari and Madhavi, 2009 have obtained formulae for the number of triangles, the cycles of smallest length and the Hamilton cycles, the cycles of longest length in Cayley graphs associated with the set of Quadratic residues modulo a prime p.The formula for the number of triangles in a Mangoldt graph M n .(Maheswari and Madhavi, 2008) is also obtained.So, it is interesting to find formula for Hamilton cycles in M n .Further one may carry out studies on these arithmetic graphs about other domination parameters li ke independent domination number, total domination number and others.
≤ n and let α 1 ,α 2 , . . .α t be the largest positive integers such that ≤ n for 1≤ i ≤ t.Then the number N of numbers ≤ n which are products of more than two distinct prime powers is given by ≤ n and for each i, let α i be the largest positive integer such that i i p α ≤ n for 1≤ i ≤ t.Among the numbers 1, 2, 3, ……, n the numbers which are powers of the prime i are ,1≤ i ≤ t and the number of the total number of numbers (among 1, 2, 3, ., n), which are powers of a single prime ≤ n is t
1 -{i, i+1} can not be an edge dominating set since the edge {i, i+1}∈E-and it is not adjacent to any of the edges in .So, E Thus E 2 is an edge dominating set.As in case 1 one can see that E 2 E Consider the graph M n -F.For any prime p j ≤ n,1≤ j ≤ t and positive integer α such that ≤ n and for any m the minimum set of edges F such γ(M n ) < γ(M n -F).This formula comes handy for large values of n for which sketching of the graph M n is unvieldy.Once the value of n is given one can calculate N and ⎡N/2⎤ gives b (M n ).
4.7.RemarkTheorem 4.4 gives the formula for bondage number of M n , namely, b(M n ) = ⎡N/2⎤ , where N is the positive integer associated with n given in Lemma 4.3.One need not draw the graph and identify