1. Introduction
In this manuscript, we are concerned with the orbit graph of some
finite group . Let be a graph, possessing a subset of
commuting elements of order two in , denoted by . The vertices of the graph
represent non-central orbits in , and two vertices have an edge
between them if they are conjugate.
Let and be the degree matrix and
adjacency matrix of the graph
respectively. Then and are the Laplacian matrix and signless Laplacian matrix
of respectively. Let be the degree of the -th vertex of , where .
The spectrum of the graph
is a multiset given by ,
where
are the eigenvalues of
having multiplicities respectively.
Similarly, the Laplacian and signless Laplacian spectrum of is given by ,
and
respectively, where are
representing the eigenvalues of having multiplicities where as
are the eigenvalues of
having multiplicities respectively.
A graph’s energy can be expressed as energy , laplacian energy , or signless laplacian energy
, depending on its
different spectra. In , Gutman
[1] defined the energy of
a graph as The concept of
laplacian energy of a graph was initially coined by Gutman and Zhou
[2] and is defined as Similarly, Cvetkovic et
al. [3], introduced the
signless laplacian energy as where and represents the set of vertices
of and set of edges of 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 ). The set
consists of all pairs of
commuting elements of that are of
the form where and 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.,
Definition 2. [14](Orbit).
Consider a group having a finite
number of elements which acts on set , and . The orbit of ,
denoted by , is the subset
defined as . In this paper, the group action is considered as a
conjugation action. Hence, the orbit is defined as
Definition 3. [15](Orbit Graph, ).
Let be a metabelian group and
be a set. An orbit graph is
denoted by and
can be defined as a graph where non-central orbits under the group
action on the set represent
vertices, i.e., , where
is a disjoint union of
distinct orbits and . In the orbit graph, two vertices and are adjacent if and are conjugate, i.e., .
Theorem 1. [15] If
is a dihedral group having
number of elements and acts on
by conjugation, then
Theorem 2. [15] Let be a quaternion group having number of elements. If acts on by conjugation, then is an empty or null
graph.
Proposition 1. [16] The number of connected components in
the graph is the multiplicity of
as the eigenvalue of .
Proposition 2. [17] The eigenvalues of the Laplacian matrix
of the complete graph, , are
with multiplicity and with multiplicity .
3. Main Results
The Laplacian energy as well as signless Laplacian energy of an orbit
graph of some dihedral groups having order and quaternion group of order have been studied in this
particular section.
Theorem 3. Let be a dihedral group having order , where is even and is odd, and let be the orbit
graph. Then .
Proof. Let be a
dihedral group having elements,
where is odd and is even, and . Then, we have:
and therefore,
Case I (Laplacian Energy): Here, we find out the
Laplacian energy of an orbit graph of the dihedral group
having order . For this, we have:
Now, the Laplacian eigenvalues of are having multiplicity and having multiplicity . Using the definition of
Laplacian energy, we get:
Case II (Signless Laplacian Energy): In this case,
we find out the signless Laplacian energy of an orbit graph of the dihedral group
having order . For this, we have:
Now, the signless Laplacian eigenvalues of are having multiplicity
and
with multiplicity . Using the
definition of signless Laplacian energy, we get:

Theorem 4. If be a
dihedral group having order ,
where and are even, and let be the orbit
graph. Then .
Proof. Consider a dihedral group having elements, where and are even, and , then, we
have:
and therefore,
Case I (a) (Laplacian Energy): Here, we find out the
Laplacian energy of an orbit graph . For this, we
have:
Now, the Laplacian eigenvalues of are having multiplicity and having multiplicity . Using the
definition of Laplacian energy, we get:
Case I (b): Here, we find out the Laplacian energy
of an orbit graph .
For this, we have:
Now, the Laplacian eigenvalues of are having multiplicity and having multiplicity . Using the
definition of Laplacian energy, we have:
Case II (a) (Signless Laplacian Energy): In this
case, we find out the signless Laplacian energy of an orbit graph of
. For this, we
have:
Now, the signless Laplacian eigenvalues of are having multiplicity
and
having multiplicity . Using the
definition of signless Laplacian energy, we get:
Case II (b): Here we find out the signless Laplacian
energy of an orbit graph of .
For this, we have:
Now, the signless Laplacian eigenvalues of are
having multiplicity and
having multiplicity . Using the
definition of signless Laplacian energy, we get:

Theorem 5. Let be a dihedral group having
order , where is odd, and let be the orbit
graph. Then .
Proof. Consider a dihedral group having order ; is odd and , then, we
have:
and therefore,
Case I (Laplacian Energy): Here, we find out the
Laplacian energy of an orbit graph of the
dihedral group of order
. For this, we have:
Now, the Laplacian eigenvalues of are having multiplicity and have multiplicity . Using the definition of Laplacian
energy, we get:
Case II (Signless Laplacian Energy): Here, we find
out the signless Laplacian energy of an orbit graph of the
dihedral group having order
. For this, we have:
Now, the signless Laplacian eigenvalues of are with multiplicity
and with multiplicity
. Using the definition of
signless Laplacian energy, we get:

Proposition 3. Let be a quaternion group of order
, and let be the orbit
graph. Then .
Proof. Since the orbit graph of the
quaternion group is a
null graph, its vertices and edges are zero. Therefore, . 
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 are
found. For even and odd, , while for
and even, ,
and for odd, . We also
find that the Laplacian energy as well as signless Laplacian energy of
an orbit graph of the quaternion group having order 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.