THE DISTANCE MATRIX AND THE DISTANCE ENERGY OF THE … · much interest. Since the fundamental...

Journal of Science and Arts Year 19, No. 4(49), pp. 823-838, 2019

ISSN: 1844 – 9581 Mathematics Section

ORIGINAL PAPER

THE DISTANCE MATRIX AND THE DISTANCE ENERGY OF THE

POWER GRAPHS OF AND

SERIFE BUYUKKOSE1, NURSAH MUTLU VARLIOGLU

1, ERCAN ALTINISIK

1

_________________________________________________

Manuscript received: 23.05.2019; Accepted paper: 11.09.2019;

Published online: 30.12.2019.

Abstract. In this study, the distance matrix and the distance energy of power graphs

on cyclic groups and dihedral groups are considered. Furthermore, some bounds for the

largerst eigenvalue of the distance matrix and the distance energy are found. Also, some

results are obtained by using these bounds.

Keywords: Power graph, distance matrix, distance energy, eigenvalue bound, energy

bound.

1. INTRODUCTION

An undirected power graph of a group is an undirected graph whose vertex set

is and two distinct vertices and are adjacent if and only if or for some

positive integer . At the first time, the concept of power graphs was introduced by Kelarev

and Quinn but they studied only directed power graphs for semigroups in [1]. A directed

power graph of a semigroup is a directed graph with vertex set and for there is an

arc from to if and only if and for some positive integer [1-3]. Then

motivated by concept of directed power graphs, Chakrabarty et al. introduced the undirected

power graph [4]. In the same paper, it was shown that the undirected power graph of

any finite group is complete if and only if is a cyclic group of order or for some

prime number and positive integer . Indeed, is always connected. Throughout this

paper, we use the brief term power graph to refer to an undirected power graph.

In this paper, we will examine the undirected power graphs of cyclic groups of

order and dihedral groups of order . For this aim, we now redefine some graph

theoretic concepts for our particular power graphs.

The distance between vertices and of a power graph denoted by ( ),

is defined to be the length of the shortest path from to . Let be the set of the identity

and generators of (say), where is Euler's function. Also,

let Then the distance matrix ( ) of the power graph is of the form

(

)

where is the identity matrix, the all-ones matrix, and ) is the distance

matrix of the power graph induced by the vertex set , i.e.,

1 Gazi University, Faculty of Sciences, Department of Mathematics, 06500 Ankara, Turkey.

E-mail: [email protected]; [email protected]; [email protected].

mailto:[email protected]



The distance matrix and… Serife Buyukkose et al.

www.josa.ro Mathematics Section

824

{ ( )

The eigenvalues of the distance matrix of are denoted by

Since ( ) is a real symmetric matrix, its eigenvalues are real and can be ordered

as

For each positive integer , the dihedral group is a non-

commutative group of order whose generators and satisfy , and

. Since , the cyclic group is a subgroup of of

order . So is a connected subgraph of The power graph , can be

considered as a copy of and copies of the complete graph which share the

identity. Moreover, the distance matrix ( ) of is of the form

( ) ( ( )

)

where

(

)

The eigenvalues of the distance matrix of are denoted by

( ) ( ) ( )

Since ( ) is a real symmetric matrix, its eigenvalues are real and can be

ordered as

( ) ( ) ( )

Analog to the definition of the graph energy [5], we can naturally define the distance

energy of the power graph on a group as the sum of the absolute values of

its distance eigenvalues ( ) ( ) ( ) i.e.,

( ) ∑| ( )|

Since calculating such graph invariants is a hard work, the bounding problem for the

largest eigenvalue of the distance matrix and the distance energy of a graph have received



825

much interest. Since the fundamental paper of Ruzieh and Powers [6] in 1990, the bounding

problem for the largest eigenvalue of the distance matrix of a graph has appeared frequently in

many papers [5, 7-10]. Furthermore, the concept of distance energy for graphs introduced by

Indual, Gutman and Vijayakumar [5]. Then, many results on lower and upper bounds for

distance energy have been obtained in [9, 11-13].

In this paper, motivated by the definition of the adjacency matrix of a power graph in

[17], the distance matrix of the power graph of a finite group are defined. Moreover, its

eigenvalues and the sum of the absolute values of its eigenvalues are called distance

eigenvalues and the distance energy of a power graph, respectively. In the following parts of

this study, sharp upper and sharp lower bounds for the largest distance eigenvalue and the

distance energy for the cyclic group and the dihedral group are obtained.

2. BOUNDS FOR THE LARGEST LAPLACIAN EIGENVALUES OF DISTANCE

MATRICES OF AND

Theorem 2.1. Let be the power graph of of order with . Then

( ) (1)

and

( ) √(( )

)

(2)

where . Moreover equality holds in (1) and (2) if and only if , for some

prime number and positive integer .

Proof. Let be the set of identity and all generators of and For the sake of

simplicity, we should label the vertices of as and as .

Then and . Now any row sum of each block and

are , and , respectively.

Let be a Perron eigenvector of ( ) corresponding to the

largest eigenvalue ( ) and let

(3)

From the eigenvalue equation, we have

( ) ( ) (4)

From (3) and the th equation of (4), we get

( ) ∑ ∑

∑ ∑



826

i.e.,

( ( ) ) (5)

Similarly, from (3), the th equation of (4) and the fact that

{ ( )}

we have

( ) ∑ ( ) ∑ ( )

{ ( )}

i.e.,

( ( ) ) (6)

From (5) and (6), we have

( ( ) ) ( ( ) )

also since and are positive, we obtain

( ( ) ) ( ( ) )

i.e.,

( )

Let

By a similar argument, using the fact that

{ } , we can show

that

( ( ) ) (7)



827

and

( ( ) ) (8)

Since and are positive and from (7) and (8), we have

( ( ) ) ( ( ) )

Thus

( ) √(( )

)

Equality holds in the lower and upper bounds for ( ) if and only if

{ ( )}

{ ( )}

Then is a complete graph and thus is a cyclic group of order for some

prime number and positive integer m. Conversely, suppose that for any prime

number and positive integer Then is complete and thus

{ ( )}

{ ( )}.

Hence, the theorem is proved.

Theorem 2.2. Let and be the power graphs of and respectively and

. Then

( ) ( ) ( ) √ (√ √ ) √

Proof. It is clear that ( ) is a principal submatrix of ( ) and from Cauchy's

Interlace Theorem in [5-6], we obtain

( ) ( )

From the definition of the distance matrix of the power graph , we have

( ) ( ( )

)

( ( )

) (

) (

)



828

( ( )

) (

) (

) (

)

where

(

) (

)

Using the relation between the principal minors of a matrix and the coefficients of its

characteristic polynomial one can obtain that the characteristic polynomial of

(

)

is and hence,

(

) √( ) (9)

Also, one can show that

(

) √ (10)

see p. 64 in [13], we have

(

)

(11)

Since

( ( )

) (

) (

) (

)

are symmetric matrices, we obtain

( ) [( ( )

) (

) (

) (

)]

( ( )

) (

) (

) (

)



829

By (9), (10) and (11), we have

( ) ( ) √ √( )

( ) √ (√ √ ) √

The proof is complete.

3. SOME UPPER AND LOWER BOUNDS FOR THE DISTANCE ENERGY OF

AND

Let be a cyclic group of order with and be its power graph. We

denote by ( ) the distance matrix of and by

( ) ( ) ( ) its eigenvalues in decreasing order. Moreover,

naturally define the distance energy ( ) of the power graph as the sum of the

absolute values of its distance eigenvalues, i.e.,

( ) ∑| ( )|

Lemma 3.1. Let be the power graph of with . Then

∑ ( )

and

∑ ( )

∑ ( )

Proof. From the definition of the trace of a matrix, we have

∑ ( )

[ ( )]

We now consider the matrix ( )

∑ ( )

[ ( ) ]



830

The th entry of is ∑ ( )

. Thus

∑ ( )

[ ( ) ]

∑ ( )

This completes the proof.

Theorem 3.2. Let ( ) be the distance energy of the power graph Then

( ) √( ∑ ( )

)

and

( ) √( ∑ ( )

)

Proof. By Lemma 3.1 and the Cauchy-Schwarz inequality, we obtain

( )

(∑| ( )|

)

∑ ( )

∑ ( )

(12)

For the proof of the first inequality, we have

( )

(∑| ( )|

)



831

∑ ( )

∑ ( )

(13)

Using (12) and (13), we obtain the required result.

Corollary 3.3. Let be a cyclic group of order for some prime number and positive

integer . Then

√ ( )

and

( ) √

Proof. We know that if is a cyclic group of order for some prime number and

positive integer then is a complete graph, and hence

(14)

By Theorem 3.2 and the equality (14), we have

√ ( )

and

( ) √

Therefore, the proof is complete.

Theorem 3.4. Let be the power graph of with . Then

( ) √ ( ∑ ( )

)

Proof. By the Cauchy-Schwarz inequality and Lemma 3.1, we obtain

( ( ) ( ))

(∑| ( )|

)



832

(∑ ( )

( ))

( ∑ ( )

( ))

and thus

( ) √ ( ∑ ( )

)


Corollary 3.5. Let be a cyclic group of order for some prime number p and positive

integer m. Then

( )

Proof. We know that if is a cyclic group of order for some prime number and

positive integer , then is a complete graph and thus ( ) . Using

Theorem 3.4, we have

( ) √ ( ∑ ( )

)

√ ( )

so the proof is completed.


( ) √( ∑ ( )

[ ( )]

)



833

Proof. By Lemma 3.1, we have

( )

(∑| ( )|

)

∑ ( )

∑| ( )|| ( )|

∑ ( )

∑| ( )|| ( )|

Since the geometric mean of nonnegative numbers is smaller than their arithmetic

mean. Thus, we have

∑ | ( )|| ( )| ∑ | ( )|| ( )| |

(∏| ( )|| ( )|

)

(∏| ( )|

)

[ ( )]

(16)

By (15) and (16) , we obtain

( ) √( ∑ ( )

[ ( )]

)


Now we consider the power graph of the dihedral group with . We

denote by ( ) the distance matrix of and by

( ) ( ) ( ) its eigenvalues in decreasing order. Moreover,

naturally define the distance enrgy ( ) of the power graph as the sum of the

absolute values of its distance eigenvalues i.e.,

( ) ∑| ( )|



834

Lemma 3.7. Let be the power graph of the dihedral group with Then

∑ ( )

and

∑ ( )

∑ ( )

Proof. From the definition of the trace of a matrix, we have

∑ ( )

[ ( )]

Now we consider the matrix

( )

( ( )

)

where is the identity matrix, the all-ones matrix and

(

)

Then, it is clear that

∑ ( )

[ ( ) ]

[ ( ) ]

By Lemma 3.1, we have

∑ ( )

∑ ( )

∑ ( )

This completes the proof.



835


√ ∑ ( )

( )

and

( ) √ [ ∑ ( )

]

Proof. By Lemma 3.7 and the Cauchy-Schwarz inequality, we obtain

( )

(∑| ( )|

)

∑ ( )

∑ ( )

[ ∑ ( )

]

For the other side of the inequaities, we have

( )

(∑| ( )|

)

∑ ( )

∑ ( )

Using (17) and (18) we get the required result.



836


( ) ( ) √ ( ∑ ( )

( )

)

Proof. By the Cauchy-Schwarz inequality and Lemma 3.7, we obtain

( ( ) ( ))

(∑| ( )|

)

(∑ ( )

( ))

( ∑ ( )

( ))

and thus

( ) ( ) √ ( ∑ ( )

( )

)


Theorem 3.10. Let and be the power graphs of and respectively with

, then

( ) ( ) ( ) √ ( √ )

Proof. Since ( ) is a principal submatrix of ( ) and from Cauchy's Interlace

Theorem, we have

( ) ( )

for Thus

( ) ( )



837

From the definition of the distance matrix of the power graph we have

( ) ( ( )

)

( ( )

) (

) (

)

( ( )

) (

) (

) (

)

where

(

) (

)

Using the relation between the principal minors of a matrix and the coefficients of its

characteristic polynomial one can obtain that the characteristic polynomial of (

) is

and hence,

∑ (

)

√( )

Also one can show that √ and are sums of the absolute values of

eigenvalues of

(

) and (

),

respectively. Since

( ( )

) (

) (

) and (

)

are symmetric matrices, we obtain

( ) ∑| ( )|

∑| ( ( )

)|

∑| (

)|

∑| (

)|

∑| (

)|



838

( ) √ ( √ )

Hence, the theorem is proved.

Conflict of interests: The authors declare that there is no conlict of interests regarding the

publication of this paper.

CONCLUSIONS

In the present paper, the distance matrix and the distance energy of power graphs on

cyclic groups and dihedral groups are considered and some bounds for the largerst eigenvalue

of the distance matrix and the distance energy are presented at the first time in the literature.

REFERENCES

[1] Kelarev, A. V., Quinn, S. J., Journal Algebra, 251, 16, 2002.

[2] Kelarev, A. V., Quinn, S. J., Contrib. General Algebra, 12, 229, 2010.

[3] Kelarev, A. V., Quinn, S. J., Comment. Math. Univ. Carolinae, 45, 1, 2004.

[4] Chakrabarty, I., Ghosh, S., Sen, M. K., Semigroup Forum, 78, 410, 2009.

[5] Indulal, G., Gutman, I., Vijaykumar, A., MATCH Commun. Math. Comput. Chem., 60,

461, 2008.

[6] Ruzieh, S. N., Powers, D. L., Linear Multilinear Algebra, 28, 75, 1990.

[7] Edelberg, M., Garey, M. R., Graham R. L., Discrete Math., 14, 23, 1976.

[8] Graham, R. L., Hoffman, A. J., Hosoya, H., Journal of Graph Theory, 1, 85, 1977.

[9] Indulal, G., Linear Algebra Appl., 430, 106, 2009.

[10] Lin, H., Shu, J., Linear Multilinear Algebra, 60, 1115, 2012.

[11] Ramane, H. S., Revankar, D. S., Gutman, I., Rao, S. B., Acharya, B. D., Walikar, H. B.,

Journal Math., 31, 59, 2008.

[12] Ramane, H. S., Revankar, D. S., Gutman, I., Rao, S. B., Acharya, B. D., Walikar, H. B.

Graph Theory Notes New York, 55, 27, 2008.

[13] Ramane, H. S., Revankar, D. S., Gutman, I., Walikar, H. B., Publ. Inst. Math., 85, 39,

2009.

[14] Chattopadhyay, S., Panigrahi, P, Atik, F, Indagationes Mathematicae, 29, 730, 2017.

[15] Horn, R. A., Johnson, C. R., Matrix analysis, Cambridge, United Kingdom:Cambridge

University Press, 42, 2012.

[16] Hwang, S. G., The American Mathematical Monthly, 111, 157, 2004.

[17] Lütkepohl, H., Handbook of matrices, Chichester, England:John Wiley & Sons, 64,

1996.

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