Color Laplacian and Color Signless Laplacian Energy of

Complement of Subgroup Graph of Dihedral Group

Abdussakir

, Mohammad Nafie Jauhari

, Nabila Umar

and Lila Aryani Puspitasari

Department of Mathematics, Universitas Islam Negeri Maulana Malik Ibrahim Malang, Malang, Indonesia

Keywords: Color Laplacian Energy, Color Signless Laplacian Energy, Subgroup Graph, Dihedral Group

Abstract: Laplacian and signless Laplacian energy of a limited graph is the most intriguing themes on territories of

energy of a chart. The new idea of energy of a graph is color energy and moreover color Laplacian and

energy on color signless Laplacian. In this paper, the formulae of energy Laplacian and color signless

Laplacian energy of supplement of subgroup charts of dihedral assemble are resolved. Color Laplacian and

color signless Laplacian range of these graph are additionally measured..

1 INTRODUCTION

The energy of a finite graph G is characterized as the

summation of total estimations of all matrix of G

eigenvalues (Balakrishnan, 2004). Gutman (1978)

introduced the concept of adjacency energy of a

graph, and now it has evolved into other concept of

energy. Researches related to the energy of a graph

have been done, such as adjacency energy (Gutman,

2001), incidence energy (Gutman et al., 2009 and

Jooyandeh et al., 2009), Harary energy (Güngör &

Çevik, 2010), Randic energy (Das et al., 2014),

maximum degree energy (Adiga & Smitha, 2009),

detour energy (Ayyaswamy & Balachandran, 2010),

matching energy (Gutman & Wagner, 2012),

distance energy (Ramane et al., 2008), covering

distance energy (Kanna et al., 2013), dominating

distance energy (Kanna et al.. 2014), Laplacian

energy (Gutman & Zhou, 2006, Zhou & Gutman,

2007 and Zhou et al.. 2008) and signless Laplacian

energy (Liu, 2010). Subsequent developments

introduced the concept of color energy of graph

(Adiga et al.. 2013) and finally color Laplacian

(Bhat & D’souza, 2015) and color signless

Laplacian energy (Bhat & D’Souza, 2017a) of

graph.

Graphs accomplished from a group have likewise

been presented, for example Cayley graph

(Heydemann, 1997), transitive Cayley graph

(Kelarev & Praeger, 2003), conjugate graph

(Erfanian & Tolue, 2012), commuting graphs

(Chelvam et al., 2011), non-commuting graphs

(Raza & Faizi, 2013), inverse graphs (Alfuraidan &

Zakariya, 2017), identity graphs (Kandasamy &

Smarandache, 2009) and subgroup graphs

(Anderson et al., 2012). Anderson et al. (2012)

defining the subgroup graph of a group G as a

directed graph containing all elements of G and two

distinct vertices x and y will be joined by an arch if

and only if xy is belong to the related subgroup.

When the given subgroup is a normal subgroup of

G, then the subgroup graph obtained is an undirected

graph and thus its complement is also an undirected

graph (Kakeri & Erfanian, 2015).

Abdussakir has determined detour energy of the

complement of subgroup graphs of dihedral group.

(2017). In this research, the formulae of color

Laplacian and color signless Laplacian spectrum and

energy of these graphs are determined.

2 LITERATURE REVIEW

Let G be a finite graph with order

and

size

. Two distinct vertices x and y are

called adjacent if they are joined by an edge in G or

xy  E(G). The adjacency matrix A(G) of G is a

matrix A(G) = [a

] of order p where a

= 1 if v



E(G) and a

= 0 if v

 E(G) (Abdussakir et al.,

2009). The degree deg(x) of a vertex x in G is the

number of vertices that adjacent with x. The degree

matrix D(G) of G is matrix D(G) = [d

] of order p

where d

= deg(v

) for i = j and d

= 0 otherwise

(Abdussakir et al., 2017).

Abdussakir, ., Jauhari, M., Umar, N. and Puspitasari, L.

Color Laplacian and Color Signless Laplacian Energy of Complement of Subgroup Graph of Dihedral Group.

DOI: 10.5220/0009926112651269

In Proceedings of the 1st International Conference on Recent Innovations (ICRI 2018), pages 1265-1269

ISBN: 978-989-758-458-9

1265

Matrix L(G) = D(G) – A(G) is called the

Laplacian matrix of graph G (Elvierayani &

Abdussakir, 2013) and matrix L

(G) = D(G) + A(G)

is called the signless Laplacian matrix of G (Ashraf

et al., 2013). The characteristic polynomial of L(G)

and L

(G) are det(L(G) - I) and det(L

(G) - 

I),

respectively. The characteristic equation roots of a

matrix are called eigenvalues of the matrix (Jog &

Kotambari, 2016). The eigenvalues of L(G) are

called Laplacian eigenvalues of G and the

eigenvalues of L

(G) are called signless Laplacian

eigenvalues of G.

Let

are Laplacian eigenvalues of

graph G. The Laplacian energy of G is defined by

Zhou & Gutman (2007) as

. (1)

In similar way, the signless Laplacian energy of

graph G is defined by Xi & Wang (2017) as

(2)

where

are signless Laplacian eigenvalues of G.

Adiga et al (2013) presented the color energy of a

graph concept motivated by the work of

Sampathkumar and Sriraj (2013). A graph G

coloring is assigning color to all vertices of G such

that two adjacent vertices have the different color

(Bondy & Murty 2008). Coloring of graph G can be

considered as a function

such that

if x and y are adjacent in G (Chartrand

et al. 2016). The minimum positive integer k is

called chromatic number of G and denoted by χ(G)

is a coloring of G (Akbari

et al., 2009).

Let G be a colored graph and c is a coloring of G.

Color matrix

of G is defined by

(3)

Eigenvalues of

are called color eigenvalues of

G and the summation of absolute values of color

eigenvalues of G is called color energy of G and

denoted by

(Adiga et al. 2013). Bhat and

D’souza (2015) and also Shigehalli and Betageri

(2015) introduced color Laplacian matrix of a

colored graph G as

Furthermore, Bhat and D’souza (2017b) defined

color signless Laplacian matrix of a colored graph G

, where D(G) is the

degree matrix of G. The eigenvalues of

are

called color Laplacian eigenvalues of G and the

eigenvalues of

are called color signless

Laplacian eigenvalues of G.

In a similar way with the definition of Laplacian

and signless Laplacian energy, the color Laplacian

energy of colored graph G of order p and size q is

defined as

(4)

and its color signless Laplacian energy is defined as

(5)

where

are color Laplacian eigenvalues and are

color signless Laplacian eigenvalues (Bhat &

D’souza, 2015, Shigehalli & Betageri, 2015 and

Bhat & D’Souza, 2017b).

If a graph G is colored with

, then color

matrix, color Laplacian matrix and color signless

Laplacian matrix of G are called chromatic matrix,

chromatic Laplacian matrix and chromatic signless

Laplacian matrix of G and denoted by

and

, respectively. Furthermore, chromatic

Laplacian energy and chromatic signless Laplacian

energy of G are denoted by

and .

Let

(s  p) are distinct color

Laplacian eigenvalues of G and

are

their multiplicities. The color Laplacian spectrum of

G is defined as

. (6)

Let

(t  p) are distinct

color Laplacian eigenvalues of G and

are their multiplicities. The color

signless Laplacian spectrum of G is defined as

. (7)

The following are previous results that will be

used in further discussion.

Theorem 2.1. (Bhat & D’souza 2015) For n  2,

then

and

Theorem 2.2. (Bhat & D’souza 2015) For n  1,

then

ICRI 2018 - International Conference Recent Innovation

1266

and

Theorem 2.3. (Bhat & D’souza 2015) The

chromatic Laplacian spectrum of

and

if m = n and n = m + 1 and

if n > m + 1.

Theorem 2.4. (Bhat & D’Souza 2017b) For n  2,

and

Theorem 2.5. (Bhat & D’Souza 2017b) For n  1,

then

and

Theorem 2.6. (Bhat & D’Souza 2017b) The

chromatic signless Laplacian spectrum of

and

Recently, graph of group has also been a research

topic discussed by many researchers. Anderson et al

(2012) presented the subgroup graph concept in a

group. Let H is any subgroup of a group G. The

subgroup graph of G is defined as a directed graph

with vertex set G such that vertex x will be adjacent

to vertex y if and only if and and

denoted by

. If H is a normal subgroup of G,

then

is an undirected simple graph and so

is an undirected simple graph too (Kakeri &

Erfanian, 2015).

Study about color Laplacian and color signless

Laplacian range and energy of subgroup charts have

not revealed yet as of recently, particularly for the

subgroup graph of dihedral gathering. Let

is the dihedral group of order 2n (n 

3). All normal subgroup of

are the subgroups

, where d is divisor of n, and , for odd n

and the subgroups

, where d is divisor of n,

, and , for even n (Abdussakir,

2017). Motivated by this condition, the color

Laplacian and color signless Laplacian spectrum and

energy of subgroup graphs of dihedral group are

studied. All subgroups discussed in this paper are

normal subgroups of

3 RESULTS

Since this paper concentrated on the colouring with

least number of colors, the primary concern of this

research are the chromatic Laplacian range and

energy and the chromatic signless Laplacian range

and energy.

Theorem 3.1. For

, then

and

Proof. By definition of subgroup graph, then the

subgroup graph

of dihedral group is a

complete graph of order 2n. By Theorem 2.1, the

proof is obtained. 

Theorem 3.2. For

, then

and

Proof. Since

is a complete graph of order

2n, then

is a null graph of order 2n. Using

Theorem 2.2 the proof is obtained. 

Theorem 3.3. For

, then

and

Proof. The subgroup graph

of dihedral

group

is an unconnected graph with two

components. The two components are complete

graphs of order n with vertex set

Color Laplacian and Color Signless Laplacian Energy of Complement of Subgroup Graph of Dihedral Group

1267

and

, respectively. Therefore,

is a complete bipartite graph . By

Theorem 2.3 and some computation, the proof is

obtained. 

Theorem 3.4. For

and n is even, then

and

Proof. The normal subgroup

of for

and n is even is

and

if and only if both i and j

is even or both i and j is odd, for 1  i, j  n – 2 and

k = 0, 1. Then, the subgroup graph

has

two kind of components and each component has

complete graph

of order n. Therefore,

is a complete bipartite graph . By

using Theorem 2.3, the proof is complete. 

Theorem 3.5. For

and n is even, then

and

Proof. The subgroup graph

has two

kind of components and each component has

complete graph

of order n. Therefore,

is a complete bipartite graph . The

proof is obvious by using Theorem 2.3. 

Theorem 3.6. For

, then

and

Proof. Since the subgroup graph

dihedral group

is a complete graph of order 2n,

by using Theorem 2.4 the proof is obtained. 

Theorem 3.7. For

, then

and

Proof. Since

is a complete graph of order

2n, we have

is a null graph of order 2n.

Using Theorem 2.5 the proof is obtained. 

Theorem 3.8. For

, then

and

Proof. Since

is a complete bipartite graph

. By Theorem 2.6 and some computations, the

desired proof is obtained. 

Theorem 3.9. For

and n is even, then

and

Proof. By the proof of Theorem 3.4. then

is a complete bipartite graph . By

using Theorem 2.5 the proof is obtained. 

Theorem 3.10. For

and n is even, then

and

Proof. Since

is a complete bipartite

graph

by the proof of Theorem 3.5, then it is

obvious by using Theorem 2.6. 

4 CONCLUSIONS

The formulae of chromatic Laplacian and chromatic

signless Laplacian spectrum and complement energy

of subgroup graphs of dihedral group for several

normal subgroups have been determined. Further

research is needed to observe chromatic Laplacian

and chromatic signless Laplacian spectrum and

complement energy of subgroup graphs of dihedral

group for the rest normal subgroups.

ACKNOWLEDGEMENT

We would like to thank Faculty of Science and

Technology, State Islamic University of Malang for

partial support to this research.

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