On the Laplacian Energy of an Orbit Graph of Finite Groups

Vijay Kumar Bhat1, Malkesh Singh1, Karnika Sharma1, Maryam Alkandari2, Latif Hanna2
1School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, Jammu and Kashmir, India
2Department of Mathematics, Kuwait University, Kuwait

Abstract

Let βH denote the orbit graph of a finite group H. Let ζ be the set of commuting elements in H with order two. An orbit graph is a simple undirected graph where non-central orbits are represented as vertices in ζ, and two vertices in ζ are connected by an edge if they are conjugate. In this article, we explore the Laplacian energy and signless Laplacian energy of orbit graphs associated with dihedral groups of order $2w$ and quaternion groups of order 2w.

Keywords: Laplacian energy, Signless laplacian energy, Orbit graph, Finite group

1. Introduction

In this manuscript, we are concerned with the orbit graph of some finite group H. Let β be a graph, possessing a subset of commuting elements of order two in H, denoted by ζ. The vertices of the graph represent non-central orbits in ζ, and two vertices have an edge between them if they are conjugate.

Let D(β) and A(β) be the degree matrix and adjacency matrix of the graph β respectively. Then L(β)=D(β)A(β) and Q(β)=D(β)+A(β) are the Laplacian matrix and signless Laplacian matrix of β respectively. Let dr be the degree of the r-th vertex of β, where r=1,2,,n.

The spectrum of the graph β is a multiset given by Spec(β)={α1j1,α2j2,,αljl}, where α1,α2,,αl are the eigenvalues of A(β) having multiplicities j1,j2,,jl respectively. Similarly, the Laplacian and signless Laplacian spectrum of β is given by LSpec(β)={η1t1,η2t2,,ηmtm}, and QSpec(β)={λ1s1,λ2s2,,λnsn} respectively, where η1,η2,,ηm are representing the eigenvalues of L(β) having multiplicities t1,t2,,tm where as λ1,λ2,,λn are the eigenvalues of Q(β) having multiplicities s1,s2,,sn respectively.

A graph’s energy can be expressed as energy ξ(β), laplacian energy L.E(β), or signless laplacian energy L.E+(β), depending on its different spectra. In 1978, Gutman [1] defined the energy of a graph as ξ(β)=αSpec(β)|α|. The concept of laplacian energy of a graph was initially coined by Gutman and Zhou [2] and is defined as L.E(β)=μLSpec(β)|μ2|E(β)||V(β)||. Similarly, Cvetkovic et al. [3], introduced the signless laplacian energy as L.E+(β)=λSpec(β)|λ2|E(β)||V(β)||, where V(β) and E(β) represents the set of vertices of β and set of edges of β respectively.

Zhou [4] studied about an upper bound for the graph’s energy in terms the number of edges, the number of vertices, and the number of zero eigenvalues. Some results about cospectral graphs and equienergetic graphs were examined by Balakrishnan [5]. In [6], Dutta et al., introduced the concept on laplacian energy for non-commuting graphs of groups having finite number of elements. Then Mahmoud et al. in [7], introduced the concept of laplacian energy on conjugacy class graph of certain finite groups. Sharma and Bhat [8] studied the orbit and conjugacy class graphs for some dihedral groups. Pirzada et al. [9] determine the Laplacian as well as signless Laplacian eigenvalues for graphs like unitary cayley graph of a commutative ring. Sharma and Bhat [10] also examined topological indices for the orbit graphs of dihedral groups. Recently, Nath et al. in [11], introduced the concept on various energies for commuting graphs of finite non-abelian groups. Das et al. [12] introduced the concepts of distance Laplacian energy as well as distance signless Laplacian energy for a connected graph Orbital regular graphs using a regular action for some solvable groups with finite number of elements has been studied by Sharma et al. [13].

The main purpose of this work is to compute the formulas for the laplacian energy and signless laplacian energy of orbit graphs of some finite groups which are dihedral groups and quaternian groups. Based on the orbits of the group’s elements, the results presented in this study revealed more group characteristics and classifications. Here, we also observe that the laplacian energy is equal to signless laplacian energy.

2. Preliminaries

Some fundamental concepts of an orbit graph that are used in this manuscript are presented in this section.

Definition 1. (The set ζ). The set ζ consists of all pairs of commuting elements of H that are of the form (g,h) where g and h are elements of the metabelian groups having a finite number of elements and the least common multiple of the order of the elements is two, i.e., ζ={(g,h)H×Hgh=hg,gh,lcm(|g|,|h|)=2}.

Definition 2. [14](Orbit). Consider a group H having a finite number of elements which acts on set ζ, and uζ. The orbit of u, denoted by O(u), is the subset defined as O(u)={huhH,uζ}. In this paper, the group action is considered as a conjugation action. Hence, the orbit is defined as O(u)={huh1hH,uζ}.

Definition 3. [15](Orbit Graph, βHζ). Let H be a metabelian group and ζ be a set. An orbit graph is denoted by βHζ and can be defined as a graph where non-central orbits under the group action on the set ζ represent vertices, i.e., |V(βHζ)|=|ζ||J|, where ζ is a disjoint union of distinct orbits and J={vH|vh=hv, hH}. In the orbit graph, two vertices x1 and x2 are adjacent if x1 and x2 are conjugate, i.e., x1=hx2.

Theorem 1. [15] If H is a dihedral group having 2w number of elements and H acts on ζ by conjugation, then βHζ={i=15Kw2i,if k is even and w2 is odd,(i=14Kw2i)(i=12Kw4i),if w and w2 are odd,Kw,if w is odd.

Theorem 2. [15] Let H be a quaternion group having 2w number of elements. If H acts on ζ by conjugation, then βHζ is an empty or null graph.

Proposition 1. [16] The number of connected components in the graph is the multiplicity of 0 as the eigenvalue of L(β).

Proposition 2. [17] The eigenvalues of the Laplacian matrix of the complete graph, Kn, are n with multiplicity n1 and 0 with multiplicity 1.

3. Main Results

The Laplacian energy as well as signless Laplacian energy of an orbit graph of some dihedral groups having order 2w and quaternion group of order 2w have been studied in this particular section.

Theorem 3. Let D2w=s,tsw=t2=1,tst1=s1 be a dihedral group having order 2w, where w is even and w2 is odd, and let βD2wζ be the orbit graph. Then L.E(βD2wζ)=L.E+(βD2wζ)=5(w2).

Proof. Let D2w be a dihedral group having 2w elements, where w2 is odd and w is even, and βD2wζ=i=15Kw2i. Then, we have:

|V(βD2wζ)|=5w2,|E(βD2wζ)|=5w2(w21)2=5w(w2)8 and therefore,

2|E(βD2wζ)||V(βD2wζ)|=w22.

Case I (Laplacian Energy): Here, we find out the Laplacian energy of an orbit graph βD2wζ=i=15Kw2i of the dihedral group D2w having order 2w. For this, we have:

LSpec(βD2wζ)={05,(w2)5(w21)}.

Now, the Laplacian eigenvalues of βD2wζ are μ1=0 having multiplicity 5 and μ2=w2 having multiplicity 5(w21). Using the definition of Laplacian energy, we get:

L.E(βD2wζ)=i=1w|μi2mw|=5|0(w22)|+5(w21)|w2w2+1|=5|(w22)|+5(w22)=2(5(w22))=10(w22)=5(w2).

Case II (Signless Laplacian Energy): In this case, we find out the signless Laplacian energy of an orbit graph βD2wζ=i=15Kw2i of the dihedral group D2w having order 2w. For this, we have:

QSpec(βD2wζ)={(w2)5,(w22)5(w21)}.

Now, the signless Laplacian eigenvalues of βD2wζ are λ1=(w2) having multiplicity 5 and λ2=(w22) with multiplicity 5(w21). Using the definition of signless Laplacian energy, we get:

L.E+(βD2wζ)=i=1w|λi2mw|=5|(w2)(w22)|+5(w21)|(w22)(w22)|=5|w2w2+1|+5(w21)|w22w2+1|=5(w21)+5(w21)=2(5(w21))=10(w21)=5(w2). ◻

Theorem 4. If D2w=s,tsw=t2=1,tst1=s1 be a dihedral group having order 2w, where w and w2 are even, and let βD2wζ be the orbit graph. Then L.E(βD2wζ)=L.E+(βD2wζ)=4(w2)(w4).

Proof. Consider a dihedral group D2w having 2w elements, where w and w2 are even, and βD2wζ=(i=14Kw2i)(i=12Kw4i), then, we have:

|V(βD2wζ)|=4w2=2w,|E(βD2wζ)|=4w2(w21)2=2w(w2)4=w(w2)2 and therefore,

2|E(βD2wζ)||V(βD2wζ)|=w22.

Case I (a) (Laplacian Energy): Here, we find out the Laplacian energy of an orbit graph βD2wζ=(i=14Kw2i). For this, we have:

LSpec(βD2wζ)={04,(w2)4(w21)}.

Now, the Laplacian eigenvalues of βD2wζ are μ3=0 having multiplicity 4 and μ4=w2 having multiplicity 4(w21). Using the definition of Laplacian energy, we get:

L.E(βD2wζ)=i=1w|μi2mw|=4|0(w22)|+4(w21)|w2(w21)|=4|(w22)|+4(w22)=2(4(w22))=8(w22)=4(w2).

Case I (b): Here, we find out the Laplacian energy of an orbit graph βD2wζ=(i=12Kw4i). For this, we have:

LSpec(βD2wζ)={02,(w4)2(w41)}.

Now, the Laplacian eigenvalues of βD2wζ are μ5=0 having multiplicity 2 and μ6=w4 having multiplicity 2(w41). Using the definition of Laplacian energy, we have:

L.E(βD2wζ)=i=1w|μi2mw|=2|0(w44)|+2(w41)|w4(w41)|=2|(w44)|+2(w44)=2(2(w44))=4(w44)=(w4).

Case II (a) (Signless Laplacian Energy): In this case, we find out the signless Laplacian energy of an orbit graph of βD2wζ=(i=14Kw2i). For this, we have:

QSpec(βD2wζ)={(w2)4,(w22)4(w21)}.

Now, the signless Laplacian eigenvalues of βD2wζ are λ2=(w2) having multiplicity 4 and λ3=(w22) having multiplicity 4(w21). Using the definition of signless Laplacian energy, we get:

L.E+(βD2wζ)=i=1t|λi2mw|=4|(w2)(w22)|+4(w21)|(w22)(w22)|=4|w2w2+1|+4(w21)|w22w2+1|=4(w21)+4(w21)=2(4(w21))=8(w21)=4(w2).

Case II (b): Here we find out the signless Laplacian energy of an orbit graph of βD2wζ=(i=12Kw4i). For this, we have:

QSpec(βD2wζ)={(w22)2,(w42)2(w41)}.

Now, the signless Laplacian eigenvalues of βD2wζ are λ5=(w22) having multiplicity 2 and λ6=(w42) having multiplicity 2(w41). Using the definition of signless Laplacian energy, we get:

L.E+(βD2wζ)=i=1w|λi2mw|=2|(w22)(w44)|+2(w41)|(w42)(w42)|=2|w22w4+1|+2(w41)|w42w4+1|=2(2(w41))=4(w41)=(w4). ◻

Theorem 5. Let D2w=s,tsw=t2=1,tst1=s1 be a dihedral group having order 2w, where w is odd, and let βD2wζ be the orbit graph. Then L.E(βD2wζ)=L.E+(βD2wζ)=2(w1).

Proof. Consider a dihedral group D2w having order 2w; w is odd and βD2wζ=Kw, then, we have:

|V(βD2wζ)|=w,|E(βD2wζ)|=w(w1)2 and therefore,

2|E(βD2wζ)||V(βD2wζ)|=w1.

Case I (Laplacian Energy): Here, we find out the Laplacian energy of an orbit graph βD2wζ=Kw of the dihedral group D2w of order 2w. For this, we have:

LSpec(βD2wζ)={01,w(w1)}.

Now, the Laplacian eigenvalues of βD2wζ are μ7=0 having multiplicity 1 and μ8=w have multiplicity (w1). Using the definition of Laplacian energy, we get:

L.E(βD2wζ)=i=1t|μi2mw|=1|0(w1)|+(w1)|w(w1)|=|(w1)|+(w1)|ww+1|=(w1)+(w1)=2(w1).

Case II (Signless Laplacian Energy): Here, we find out the signless Laplacian energy of an orbit graph βD2wζ=Kw of the dihedral group D2w having order 2w. For this, we have:

QSpec(βD2wζ)={(2w2)1,(w2)(w1)}.

Now, the signless Laplacian eigenvalues of βD2wζ are λ7=(2w2) with multiplicity 1 and λ8=(w2) with multiplicity (w1). Using the definition of signless Laplacian energy, we get:

L.E+(βD2wζ)=i=1w|λi2mw|=|(2w2)(w1)|+(w1)|(w2)(w1)|=|(2w2w+1)+(w1)|+(w1)|1|=(w1)+(w1)=2(w1). ◻

Proposition 3. Let Q2w be a quaternion group of order 2w, and let βQ2wζ be the orbit graph. Then L.E(βQ2wζ)=L.E+(βQ2wζ)=0.

Proof. Since the orbit graph βQ2wζ of the quaternion group Q2w is a null graph, its vertices and edges are zero. Therefore, L.E(βQ2wζ)=L.E+(βQ2wζ)=0◻

4. Conclusion

In this manuscript, the general formula for the Laplacian energy and the general formula for signless Laplacian energy for the orbit graph of dihedral groups having order 2w are found. For w even and w2 odd, L.E(βD2wζ)=L.E+(βD2wζ)=5(w2), while for w and w2 even, L.E(βD2wζ)=L.E+(βD2wζ)=4(w2)(w4), and for w odd, L.E(βD2wζ)=L.E+(βD2wζ)=2(w1). We also find that the Laplacian energy as well as signless Laplacian energy of an orbit graph of the quaternion group having order 2w are zero.

Conflict of Interest

The authors declare no conflict of interest.

Funding

Malkesh Singh is receiving funds from SMVDU via Grant/Award Number: SMVDU/R&D/21/5121-5125 for this research.

References:

  1. Gutman, I., 1978. The Energy of Graph. Der. Math. stat. Sekt. Forschungszent Graz, 103, pp.1-22.
  2. Gutman, I. and Zhou, B., 2006. Laplacian energy of a graph. Linear Algebra Appl., 414, pp.29-37.
  3. Cvetkovic, D., Rowlinson, P. and Simic, S. K., 2007. Signless Laplacian of Finite Graphs. Linear Algebra Appl., 42(1), pp.155-171.
  4. Zhou, B., 2004. On the energy of a graph. Kragujevac J. Sci., 26, pp.5-12.
  5. Balakrishnan, R., 2004. The energy of a graph. Linear Algebra Appl., 387, pp.287-295.
  6. Dutta, P. and Nath, R. K., 2018. On Laplacian Energy of Non Commuting Graphs of Finite Groups. J. linear topol. algeb, 7(2), pp.121-132.
  7. Mahmoud, R., Fadzil, A. F. A., Sarmin, N. H. and Erfanian, A., 2019. On Laplacian Energy of Conjugacy Class Graph of Some Finite Groups. Matematika, 35(1), 59-65.
  8. Sharma, K. and Bhat, V. K., 2022. On the Orbits of some Metabelian Groups. J. App. and Eng. Math., 12(3), pp.799-807.
  9. Pirzada, S., Barati, Z. and Afkhami, M., 2021. On Laplacian spectrum of unitary Cayley graphs. Acta Universitatis Sapientiae, Informatica, 13(2), pp.251-264.
  10. Sharma, K. and Bhat, V. K., 2023. On Some Topological Indices for the Orbit Graph of Dihedral Groups. J. Comb. Math. Comb. Comput., 117, pp.95-208.
  11. Nath, R. k., Dutta, P. and Bagghi, B., 2020. Various Energies of Commuting Graphs of Finite Non Abelian Groups. Khayyam J. Math, 6(1), pp.27-45.
  12. Das, K. C., Aouchiche, M. and Hansen, P., 2018. On (distance) Laplacian energy and (distance) signless Laplacian energy of graphs. Discret. Appl. Math., 243, pp.172-185.
  13. Sharma, K., Bhat, V. K. and Singh, P., 2023. On the Orbital Regular Graph of Finite Solvable Groups. Util. Math., 118, pp.15-25.
  14. Goodman F. M., 2003. Algebra Abstract and Concrete Streesing Symmetry, Prentice Hall, 2nd edition.
  15. Omer, S. M. S., Sarmin, N. H. and Erfanian, A., 2015. The orbit graph for non abelian group. Int. J. Pure Appl. Math. Sci., 102(4), pp.747-755.
  16. Beineke, L. W. and Wilson, R. J., 2007. Topics in Algebraic Graph Theory, Combinatorics, Probability and Computing. United States of America: Cambridge University Press.
  17. Bapat, R. B., 2010. Graphs and Matrices. New York: Springer.