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Randić energy was first defined in the paper [1]. Using minimum covering set, we have introduced the minimum covering Randić energy
*RE _{C}* (

*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.

Study on energy of graphs goes back to the year 1978, when I. Gutman [

Theories on the mathematical concepts of graph energy can be seen in the reviews [

It was in the year 1975, Milan Randić invented a molecular structure descriptor called Randić index which is defined as [

Motivated by this S.B. Bozkurt et al. [

The characteristic equation of

Further studies on Randić energy can be seen in the papers [

In the year 2012 C Adiga et al. [

The minimum covering eigenvalues of the graph G are roots of the characteristic equation

Results 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

where

The characteristic polynomial of

Example 1: i)

i)

Minimum covering Randić eigenvalues are

Minimum covering Randić energy,

ii)

Minimum covering Randić eigenvalues are

Minimum covering Randić energy,

Therefore minimum covering Randić energy depends on the covering set.

Theorem 2.1 For

Proof. Let

Characteristic polynomial is

Characteristic equation is

Minimum covering Randić Spec

Minimum covering Randić energy,

Definition 2.1 Thorn graph of

Theorem 2.2 For

graph

Proof.

Characteristic polynomial is

Characteristic equation is

Minimum covering Randić Spec

Minimum covering Randić energy is,

Definition 2.2 Cocktail party graph is denoted by

Theorem 2.3 The minimum covering Randić energy,

Proof. Consider cocktail party graph

Characteristic polynomial is,

Characteristic equation is,

Minimum covering Randić Spec

Minimum covering Randić energy,

Theorem 2.4 For

Proof. Let

Characteristic equation is

Minimum covering Randić Spec

Minimum covering Randić energy,

Definition 2.3 Crown graph

Theorem 2.5 For

Proof. For the crown graph

Characteristic polynomial is

Characteristic equation is

Minimum covering Randić Spec

Minimum covering Randić energy,

Theorem 2.6 The minimum covering Randić energy,

Proof. For the complete bipartite graph

Characteristic equation is

Minimum covering Randić Spec

Minimum covering Randić energy,

Definition 2.4 Friendship graph is the graph obtained by taking n copies of the cycle graph

Theorem 2.7 The minimum covering Randić energy,

Proof. For a friendship graph

Characteristic equation is

Minimum covering Randić Spec

Minimum covering Randić energy,

Theorem 2.8 Let G be a graph with vertex set

Proof. i) We know that the sum of the eigenvalues of

ii) Similarly the sum of squares of the eigenvalues of

Mclelland’s [

Theorem 2.9 Let G be a simple graph with n vertices and m edges . If C is the minimum covering set and

Proof.

Canchy Schwarz inequality is

If

Since arithmetic mean is greater than or equal to geometric mean we have

Now consider,

Theorem 2.10 If

Proof. For any nonzero vector X, we have by [

Just like Koolen and Moulton’s [

Theorem 2.11 If G is a graph with n vertices and m edges and

Proof.

Cauchy-Schwartzin equality is

Put

Let

For decreasing function

Since

Milovanović [

Theorem 2.12 Let G be a graph with n vertices and m edges. Let

Proof. For real numbers

If

But

Theorem 2.13 Let G be a graph with n vertices and m edges. Let

Proof. Let

Put

The question of when does the graph energy becomes a rational number was answered by Bapat and S. pati in their paper [

Theorem 2.14 Let G be a graph with a minimum covering set C. If the minimum covering Randić energy

Proof. Proof is similar to theorem 3.7 of [

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. 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 com- plete graph, crown graph, complete bipartite graph, cocktail party graph and friendship graphs are also computed.

The authors are thankful to anonymous referees for their valuable comments and useful suggestions.

Both the authors worked together for the preparation of the manuscript and both of us take the full responsibility for the content of the paper. However second author typed the paper and both of us read and approved the final manuscript.

The authors hereby declares that there are no issues regarding the publication of this paper.

Kanna, M.R.R. and Jagadeesh, R. (2016) Minimum Covering Randić Energy of a Graph. Advances in Linear Algebra & Matrix Theory, 6, 116-131. http://dx.doi.org/10.4236/alamt.2016.64012