1 Introduction

Randić energy of specific graphs

Saeid Alikhani111Corresponding author. E-mail: alikhani@yazd.ac.irand Nima Ghanbari

Department of Mathematics, Yazd University

89195-741, Yazd, Iran

Abstract

Let be a simple graph with vertex set . The Randić matrix of , denoted by , is defined as the matrix whose -entry is if and are adjacent and for another cases. Let the eigenvalues of the Randić matrix be which are the roots of the Randić characteristic polynomial . The Randić energy of is the sum of absolute values of the eigenvalues of . In this paper we compute the Randić characteristic polynomial and the Randić energy for specific graphs .

Mathematics Subject Classification: 15A18, 05C50.
Keywords: Randić matrix; Randić energy; Randicć characteristic polynomial; eigenvalues.

## 1 Introduction

In this paper we are concerned with simple finite graphs, without directed, multiple, or weighted edges, and without self-loops. Let be such a graph, with vertex set . If two vertices and of are adjacent, then we use the notation . For , the degree of the vertex , denoted by , is the number of the vertices adjacent to .

Let be adjacency matrix of and its eigenvalues. These are said to be be the eigenvalues of the graph and to form its spectrum [4]. The energy of the graph is defined as the sum of the absolute values of its eigenvalues

 E(G)=n∑i=1|λi|.

Details and more information on graph energy can be found in [6, 7, 9, 10].

In 1975 Milan Randić invented a molecular structure descriptor defined as [11]

 R(G)=∑vi∼vj1√didj.

The Randić-index-concept suggests that it is purposeful to associate to the graph a symmetric square matrix . The Randić matrix is defined as [2, 3, 8]

 rij=⎧⎪⎨⎪⎩1√didjif vi∼vj0otherwise.

Denote the eigenvalues of the Randić matrix by and label them in non-increasing order. Similar to characteristic polynomial of a matrix, we consider the Randić characteristic polynomial of (or a graph ), as which is equal to . The Randić energy [2, 3, 8] of is defined as

 RE(G)=n∑i=1|ρi|.

For several lower and upper bounds on Randić energy, see [2, 3, 8].

In Section 2, we obtain the Randić characteristic polynomial and energy of specific graphs. As a result, we show that for every natural number , there exists a graph such that . In Section 3, we find Randić energy of specific graphs with one edge deleted.

As usual we denote by a matrix all of whose entries are equal to .

## 2 Randić characteristic polynomial and Randić energy of specific graphs

In this section we study the Randić characteristic polynomial and the Randić energy for certain graphs. The following theorem gives a relationship between the Randić energy and energy of path .

###### Lemma 1

[8] Let be the path on vertices. Then

 RE(Pn)=2+12E(Pn−2).

The following theorem gives the Randić energy of even cycles.

###### Lemma 2

[12] Let be the cycle on vertices for . Then

 RE(C2n)=2sin((⌊n2⌋+12)πn)sinπ2n.

Here we shall compute the Randić characteristic polynomial of paths and cycles.

###### Theorem 1

. For , the Randić characteristic polynomial of the path graph satisfy

 RP(Pn,λ)=(λ2−1)(λΛn−3−14Λn−4),

where for every , with and .

Proof. For every , consider

 Bk:=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝λ−1200…000−12λ−120…0000−12λ−12…00000−12λ…000⋮⋮⋮⋮⋱⋮⋮⋮0000…λ−1200000…−12λ−120000…0−12λ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠k×k,

and let . It is easy to see that .

Suppose that . We have

 RP(Pn,λ)=det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝λ−1√20…000−1√2000⋮Bn−1⋮000−1√2000…0−1√2λ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠n×n.

So

 RP(Pn,λ)=λdet⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0⋮Bn−20−1√20…0−1√2λ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠+1√2det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−1√2−12…0000⋮Bn−3⋮0−1√200…−1√2λ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

And so

 RP(Pn,λ)=λ(λΛn−2+1√2det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0Bn−3⋮−120…0−1√2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠)−12det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0Bn−3⋮−1√20…−1√2λ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

Therefore

 RP(Pn,λ)=λ2Λn−2−12λΛn−3−12λΛn−3−√24det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0Bn−4⋮−120…0−1√2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,

and so

 RP(Pn,λ)=λ2Λn−2−λΛn−3+14Λn−4=λ2(λΛn−3−14Λn−4)−λΛn−3+14Λn−4.

Hence

 RP(Pn,λ)=(λ2−1)(λΛn−3−14Λn−4).\vbox{\hrule width 100% height 1px\hbox{\vrule w% idth 1px\kern 0.0pt\vbox{\kern 0.0pt\vbox to 5.690551pt{\hfill}\kern 0.0pt}% \kern 0.0pt\vrule width 1px}\hrule width 100% height 1px}\vskip12.0ptplus4.0ptminus4.0pt
###### Theorem 2

. For , the Randić characteristic polynomial of the cycle graph is

 RP(Cn,λ)=λΛn−1−12Λn−2−(12)n−1,

where for every , with and .

Proof. Similar to the proof of Theorem 1, for every , we consider

 Bk:=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝λ−1200…000−12λ−120…0000−12λ−12…00000−12λ…000⋮⋮⋮⋮⋱⋮⋮⋮0000…λ−1200000…−12λ−120000…0−12λ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠k×k,

and let . We have .

Suppose that . We have

 RP(Cn,λ)=det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝λ−120…0−12−120⋮Bn−10−12⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠n×n.

So

 RP(Cn,λ)=λΛn−1+12det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−12−12…00⋮Bn−2−12⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠+(−1)n+1(−12)det⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−12⋮Bn−20−120…−12⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

And so,

Therefore,

 RP(Cn,λ)=λΛn−1−14Λn−2+(−1)n(−12)n−1(12)+(−1)n+1(−12)n+14(−1)2n+1Λn−2.

Hence

 RP(Cn,λ)=λΛn−1−12Λn−2−(12)n−1.\vbox{\hrule width 100% height 1px\hbox{% \vrule width 1px\kern 0.0pt\vbox{\kern 0.0pt\vbox to 5.690551pt{\hfill}\kern 0% .0pt}\kern 0.0pt\vrule width 1px}\hrule width 100% height 1px}\vskip12.0ptplus4.0ptminus4.0pt

In [12] has shown that the Randić energy of , the star on vertices and the complete bipartite graph is . Here using the Randić characteristic polynomial, we prove these results. We need the following lemma:

###### Lemma 3

. [4] If is a nonsingular square matrix, then

 det(MNPQ)=det(M)det(Q−PM−1N).
###### Theorem 3

. For ,

• The Randić characteristic polynomial of the star graph is

 RP(Sn,λ)=λn−2(λ2−1).
• The Randić energy of is

 RE(Sn)=2.

Proof.

1. It is easy to see that the Randić matrix of is . We have

 det(λI−R(Sn))=det⎛⎜⎝λ−1√n−1J1×(n−1)−1√n−1J(n−1)×1λIn−1⎞⎟⎠.

Using Lemma 3,

 det(λI−R(Sn))=λdet(λIn−1−1√n−1J(n−1)×11λ1√n−1J1×(n−1)).

We know that . Therefore

 det(λI−R(Sn))=λdet(λIn−1−1λ(n−1)Jn−1)=λ2−ndet(λ2In−1−1n−1Jn−1).

Since the eigenvalues of are (once) and 0 ( times), the eigenvalues of are (once) and 0 ( times). Hence

 RP(Sn,λ)=λn−2(λ2−1).
2. It follows from Part (i).

###### Theorem 4

. For ,

• the Randić characteristic polynomial of complete graph is

 RP(Kn,λ)=(λ−1)(λ+1n−1)n−1.
• the Randić energy of is

 RE(Kn)=2.

Proof.

1. It is easy to see that the Randić matrix of is . Therefore

 RP(Kn,λ)=det(λI−1n−1J+1n−1I)=det((λ+1n−1)I−1n−1J).

Since the eigenvalues of are (once) and 0 ( times), the eigenvalues of are (once) and 0 ( times). Hence

 RP(Kn,λ)=(λ−1)(λ+1n−1)n−1.
2. It follows from Part (i).

###### Theorem 5

. For natural number ,

• The Randić characteristic polynomial of complete bipartite graph is

 RP(Km,n,λ)=λm+n−2(λ2−1).
• The Randić energy of is

 RE(Km,n)=2.

Proof.

1. It ie easy to see that the Randić matrix of is . Using Lemma 3 we have

 det(λI−R(Km,n))=det⎛⎜⎝λIm−1√mnJm×n−1√mnJn×mλIn⎞⎟⎠.

So

 det(λI−R(Km,n))=det(λIm)det(λIn−1√mnJn×m1λIm1√mnJm×n).

We know that . Therefore

 det(λI−R(Km,n))=λmdet(λIn−1λnJn)=λm−ndet(λ2In−1nJn).

The eigenvalues of are (once) and 0 ( times). So the eigenvalues of are (once) and 0 ( times). Hence

 RP(Km,n,λ)=λm+n−2(λ2−1).
2. It follows from Part (i).

Let be any positive integer and be friendship graph with vertices and edges. In other words, the friendship graph is a graph that can be constructed by coalescence copies of the cycle graph of length with a common vertex. The Friendship Theorem of Erdős, Rényi and Sós [5], states that graphs with the property that every two vertices have exactly one neighbour in common are exactly the friendship graphs. The Figure 1 shows some examples of friendship graphs. Here we shall investigate the Randić energy of friendship graphs.

###### Theorem 6

. For ,

• The Randić characteristic polynomial of friendship graph is

 RP(Fn,λ)=(λ2−14)n−1(λ−1)(λ+12)2.
• The Randić energy of is

 RE(Fn)=n+1.

Proof.

1. The Randić matrix of is

 R(Fn)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝012√n12√n⋯12√n12√n12√n012…0012√n120…00⋮⋮⋮⋱⋮12√n00…01212√n00…120⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠(2n+1)×(2n+1).

Now for computing , we consider its first row. The cofactor of the first array in this row is

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝λ−12…00−12λ…00⋮⋮⋱⋮⋮00…λ−1200…−12λ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠

and the cofactor of another arrays in the first row are similar to

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−12√n−12…00−12√nλ…00⋮⋮⋱⋮⋮−12√n0…λ−12−12√n0…−12λ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠

Now, by straightforward computation we have the result.

2. It follows from Part (i).

Remark. In [1] has shown that the energy of a graph cannot be an odd integer. Since for , the Randić energy can be odd or even integer. More precisely we have:

###### Corollary 1

. For every natural number , there exists a graph such that .

Proof. If then consider and for we consider friendship graphs .

Let be any positive integer and be Dutch Windmill Graph with vertices and edges. In other words, the graph is a graph that can be constructed by coalescence copies of the cycle graph of length with a common vertex. The Figure 2 shows some examples of Dutch Windmill graphs. Here we shall investigate the Randić energy of Dutch Windmill graphs.

###### Theorem 7

. For ,

• The Randić characteristic polynomial of friendship graph is

 RP(Dn4,λ)=λn+1(λ2−12)n−1(λ2−1).
• The Randić energy of is

 RE(Dn4)=2+(n−1)√2.

Proof.

1. The Randić matrix of is

 R(Dn4)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝012√n12√n0⋯12√n12√n012√n0012…00012√n0012…000012120…000⋮⋮⋮⋮⋱⋮⋮12√n000…001212√n000…00120000…12120⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠(3n+1)×(3n+1).

Let , and .

Then

Now, by the straightforward computation we have the result.

2. It follows from Part (i).

## 3 Randić energy of specific graphs with one edge deleted

In this section we obtain the randić energy for certain graphs with one edge deleted. We need the following lemma:

###### Lemma 4

. Let . Then

 RE(G)=Re(G1)+RE(G2)+…+RE(Gm).

Here we state the following easy results:

###### Lemma 5

.

1. If , then , where .

2. If , (), then

3. Let be the star on vertices and . Then for any ,

 RE(Sn−e)=RE(Sn−1)=2.
###### Theorem 8

. For ,

• Let be an edge of complete graph . The Randić characteristic polynomial of is

 RP(Kn−e,λ)=λ(λ−1)(λ+2n−1)(λ+1n−1)n−3.
• The Randić energy of is

 RE(Kn−e)=2.

Proof.

1. The Randić matrix of is

 ⎛⎜⎝02×21√(n−1)(n−2)J2×(n−2)1√(n−1)(n−2)J(n−2)×21n−1(J−I)n−2⎞⎟⎠.

Therefore

 det(λI−R(Kn−e))=det⎛⎜ ⎜⎝λI2−1√(n−1)(n−2)J2×(n−2)−1√(n−1)(n−2)J(n−2)×2λn−1(J−I)n−2⎞⎟ ⎟⎠.

Similar to the proof of Theorem 5(i) we have the result.

2. It follows from Part (i).

###### Theorem 9

. For ,

• The Randić characteristic polynomial of complete bipartite graph is

 RP(Km,n−e,λ)=λm+n−4(λ2−1)(λ2−1mn).
• The Randić energy of is

 RE(Km,n−e)=2+2√mn.

Proof.

1. The Randić matrix of is

 (0m×mAAt0n×n),

where .So

 det(λI−R(Km,n−e))=λImdet(λIn−At1λImA).

Similar to the proof of Theorem 5(i) we have the result.

2. It follows from Part (i).

## References

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