Minimum Covering Randić Energy of a Graph

Randić energy was first defined in the paper [1]. Using minimum covering set, we have introduced the minimum covering Randić energy ( ) C RE G of a graph G in this paper. This paper contains computation of minimum covering Randić energies for some standard graphs like star graph, complete graph, thorn graph of complete graph, crown graph, complete bipartite graph, cocktail graph and friendship graphs. At the end of this paper, upper and lower bounds for minimum covering Randić energy are also presented.


Introduction
Study on energy of graphs goes back to the year 1978, when I. Gutman [2] defined this while working with energies of conjugated hydrocarbon containing carbon atoms.All graphs considered in this paper are assumed to be simple without loops and multiple edges.Let .
Theories on the mathematical concepts of graph energy can be seen in the reviews [3], papers [4] [5] [6] and the references cited there in.For various upper and lower bounds for energy of a graph can be found in papers [7] [8] and it was observed that graph energy has chemical applications in the molecular orbital theory of conjugated mo-lecules [9] [10].

Randić Energy
It was in the year 1975, Milan Randić invented a molecular structure descriptor called Randić index which is defined as [11] ( ) ( ) Motivated by this S.B. Bozkurt et al. [1] defined Randić matrix and Randić energy as follows.Let G be graph of order n with vertex set  and edge set E.
Randić matrix of G is a n n × symmetric matrix defined by ( ) ( ) , where ( ) The characteristic equation of ( ) The roots of this equation is called Randić eigenvalues of G. Since ( ) R G is real and symmetric, its eigenvalues are real numbers and we label them in decreasing order Randić energy of G is defined as ( ) .
Further studies on Randić energy can be seen in the papers [12] [13] [14] and the references cited there in.

Minimum Covering Energy
In the year 2012 C Adiga et al. [15] introduced minimum covering energy of a graph, which depends on its particular minimum cover.A subset C of vertex set V is called a covering set of G if every edge of G is incident to at least one vertex of C. Any covering set with minimum cardinality is called a minimum covering set.If C is a minimum covering set of a graph G then the minimum covering matrix of G is the n n × matrix defined by ( ) ( ) , where ( ) The minimum covering eigenvalues of the graph G are roots of the characteristic equation A G is real and symmetric, its eigenvalues are real numbers and we label them in the order The minimum covering energy of G is defined as ( ) .

Properties of Minimum Covering Randić Eigenvalues
Theorem 2.8 Let G be a graph with vertex set ii) Similarly the sum of squares of the eigenvalues of 2 , where .

Bounds for Minimum Covering Randić Energy
Mclelland's [8] gave upper and lower bounds for ordinary energy of a graph.Similar bounds for

( )
C RE G are given in the following theorem.
Theorem 2.9 Let G be a simple graph with n vertices and m edges .If C is the minimum covering set and ( ) Canchy Schwarz inequality is .
Since arithmetic mean is greater than or equal to geometric mean we have ( ) .
Now consider, ( ) Proof.For any nonzero vector X, we have by [16], ( ) where J is a unit column matrix.
Just like Koolen and Moulton's [17] upper bound for energy of a graph, an upper bound for ( ) C RE G is given in the following theorem.Theorem 2.11 If G is a graph with n vertices and m edges and Milovanović [18] bounds for minimum covering Randić energy of a graph are given in the following theorem.Theorem 2.12 Let G be a graph with n vertices and m edges.Let and [ ] x denotes the integral part of a real number.
Proof.For real numbers

∑
The question of when does the graph energy becomes a rational number was answered by Bapat and S. pati in their paper [21].Similar result for minimum covering Randić energy is obtained in the following theorem.Proof.Proof is similar to theorem 3.7 of [15].

Conclusion
It was proved in this paper that the minimum covering Randić energy of a graph G depends on the covering set that we take for consideration.

2 ,
matrix of the graph G with its eigenvalues 1 assumed in decreasing order.Since A is real symmetric, the eigenvalues of G are real numbers whose sum equal to zero.The sum of the absolute eigenvalues values of G is called the energy ( ) E G of G. i.e.,

Theorem 2 .
13 Let G be a graph with n vertices and m edges.Let

Theorem 2 .
14 Let G be a graph with a minimum covering set C. If the minimum covering Randić energy C is a minimum covering set of a graph G then the minimum covering Randić matrix of G is the n n 1.3.Minimum Covering Randić EnergyResults on Randić energy and minimum covering energy of graph G motivates us to define minimum covering Randić energy.Consider a graph G with vertex set If

2. Main Results and Discussion 2.1. Minimum Covering Randić Energy of Some Standard Graphs
Therefore minimum covering Randić energy depends on the covering set.
C RE G ≈ 4Friendship graph is the graph obtained by taking n copies of the cycle graph 3C with a vertex in common.It is denoted by 3 Upper and lower bounds for minimum covering Randić energy are established.A generalized expression for minimum covering Randić energies for star graph, complete graph, thorn graph of complete graph, crown graph, complete bipartite graph, cocktail party graph and friendship graphs are also computed.