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].
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
The distance matrix and… Serife Buyukkose et al.
ISSN: 1844 – 9581 Mathematics Section
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
( ) ∑ ∑
∑ ∑
The distance matrix and… Serife Buyukkose et al.
www.josa.ro Mathematics Section
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)
The distance matrix and… Serife Buyukkose et al.
ISSN: 1844 – 9581 Mathematics Section
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
( ) ( ( )
)
( ( )
) (
) (
)
The distance matrix and… Serife Buyukkose et al.
www.josa.ro Mathematics Section
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
( ) [( ( )
) (
) (
) (
)]
( ( )
) (
) (
) (
)
The distance matrix and… Serife Buyukkose et al.
ISSN: 1844 – 9581 Mathematics Section
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 ( )
∑ ( )
[ ( ) ]
The distance matrix and… Serife Buyukkose et al.
www.josa.ro Mathematics Section
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
( )
(∑| ( )|
)
The distance matrix and… Serife Buyukkose et al.
ISSN: 1844 – 9581 Mathematics Section
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
( ( ) ( ))
(∑| ( )|
)
The distance matrix and… Serife Buyukkose et al.
www.josa.ro Mathematics Section
832
(∑ ( )
( ))
( ∑ ( )
( ))
and thus
( ) √ ( ∑ ( )
)
The proof is complete.
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.
Theorem 3.6. Let be the power graph of with . Then
( ) √( ∑ ( )
[ ( )]
)
The distance matrix and… Serife Buyukkose et al.
ISSN: 1844 – 9581 Mathematics Section
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
( ) √( ∑ ( )
[ ( )]
)
The proof is complete.
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.,
( ) ∑| ( )|
The distance matrix and… Serife Buyukkose et al.
www.josa.ro Mathematics Section
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.
The distance matrix and… Serife Buyukkose et al.
ISSN: 1844 – 9581 Mathematics Section
835
Theorem 3.8. Let be the power graph of with . Then
√ ∑ ( )
( )
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.
The distance matrix and… Serife Buyukkose et al.
www.josa.ro Mathematics Section
836
Theorem 3.9. Let be the power graph of with . Then
( ) ( ) √ ( ∑ ( )
( )
)
Proof. By the Cauchy-Schwarz inequality and Lemma 3.7, we obtain
( ( ) ( ))
(∑| ( )|
)
(∑ ( )
( ))
( ∑ ( )
( ))
and thus
( ) ( ) √ ( ∑ ( )
( )
)
The proof is complete.
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
( ) ( )
The distance matrix and… Serife Buyukkose et al.
ISSN: 1844 – 9581 Mathematics Section
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
( ) ∑| ( )|
∑| ( ( )
)|
∑| (
)|
∑| (
)|
∑| (
)|
The distance matrix and… Serife Buyukkose et al.
www.josa.ro Mathematics Section
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.