J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
1Ae8.
On Sombor Energy of the Nilpotent Graph of the Ring of Integers Modulo A Lalu Riski Wirendra Putra1 .
Jimboy R.
Albaracin2 .
I Gede Adhitya Wisnu Wardhana3O Department of Mathematics.
University of Mataram.
Indonesia.
g1d021027@student.
id, 3 adhitya.
wardhana@unram.
Mathematics Program.
College of Science.
University of The Philippines Cebu.
Philippines, jralbaracin@up.
Abstract.
In chemical graph theory, chemical compounds are represented as graphs where atoms are represented as vertices, and the bonds connecting the atoms are represented as edges.
In 2021.
Gowtham and Swamy discovered another type of graph energy, called the Sombor energy.
This discovery was motivated by GutmanAos introduction of the Sombor index in the same year.
In the field of abstract algebra, rings can also be represented as graphs.
In this article, we aim to explore the Sombor energy of some nilpotent graphs of rings, particularly the ring of integers modulo A.
Key words and Phrases: Sombor matrix, characteristic polynomial, energy, graph, ring of integer modulo.
INTRODUCTION
Mathematical graph theory is used to represent chemical compounds, namely chemical graph theory.
In chemical graph theory, chemical compounds are modelled as graphs where the atoms that make up the compound are represented as vertices.
Each vertex corresponds to an atom.
The bonds, which are the connections between atoms, are represented as edges.
These representations help in visualizing the structure of molecules, analyzing their properties, and predicting the behaviour of chemical compounds.
By understanding the graph representation, chemists can gain insights into the molecular structure and interactions within compounds, facilitating the study of their reactivity, stability, and various physical and chemical This formalism provides a systematic and mathematical framework for O Corresponding author 2020 Mathematics Subject Classification: 05E16, 05C90, 20C05.
Received: 11-01-2025, accepted: 31-05-2025.
the depiction and analysis of chemical structures, facilitating a comprehensive understanding of the relationships between atoms and bonds within a given compound .
Research on graphs representing groups or rings is an emerging trend that bridges algebraic structures with graph theory.
This interdisciplinary field explores various types of graphs associated with algebraic entities and their applications.
Among these, the coprime graph of a group is a notable structure, where vertices represent group elements, and an edge connects two vertices if the orders of the elements are coprime .
Conversely, the noncoprime graph focuses on elements whose orders are not coprime .
This field also includes studies on numerical invariants and connectivity indices.
Researchers study topological indices as numerical invariants that characterize graph structures, such as the distance-based indices like Wiener index .
um of vertex pair distance.
and degree-based indices like the RandicA and harmonic indices, which highlight degree distributions.
These indices reveal connectivity and complexity, offering insights into algebraic structures while enabling classification and analysis of groups and rings.
Applications span cryptography, chemistry, and combinatorics, making this a dynamic research field .
, .
In this paper, we contribute to this area by exploring another related topic:
the energy of a simple graph.
The energy of a simple graph is a numerical invariant derived from the eigenvalues of its matrix representation for example adjacent It is defined as the sum of the absolute values of all its eigenvalues .
Graph energy has applications in various fields, including chemistry, where it is used to model molecular stability, and in network analysis for assessing structural properties and connectivity .
Specifically, we give some properties of the such type of graph representation of the ring of integers modulo A, namely nilpotent graph which is denoted as ZA , concerning its Sombor energy.
A nilpotent graph of a ring is a simple graph where the vertices are all the elements of the ring, and two elements are said to be adjacent if and only if the multiplications is a nilpotent element .
Previous studies give some important properties of the nilpotent graph for the ring ZA where A is a prime number or a power of 2.
Lemma 1.
The nilpotent graph of the ring of integer modulo ZA where A is a prime number is a star graph K1,AOe1 .
And for the case with a different order of elements or operations, the result can be seen in the following Lemma.
Lemma 1.
If ZA is the ring of integers modulo A where A = 2k for some k OO N, the nilpotent graph of ZA , denoted as eZA , is contains 2kOe1 star subgraphs K1,2kOe1 .
These two lemmas have nilpotent vertices as the center of every star graph .
and are important to find the degree of the vertices in the graph.
The degree of vertices is used to formulate the Sombor matrix and the energy graph of the nilpotent graph of the ring.
Definition 1.
3 (Degree of a Verte.
The degree of v OO V .
denoted by dv is defined as the number of vertices adjacent to v where e is a simple graph and V .
is the set of all its vertices.
Based on Lemma 1.
1 and 1.
2 the vertices in the graph have the degree determined by the structural properties of the graph, as described in detail in the following theorem.
Theorem 1.
If eZA is the nilpotent graph of ZA where A is a prime number, then d0 = A Oe 1 and dv = 1 for all v = 1, 2, .
A Oe 1.
Proof.
By Lemma 1.
1, the graph eZA is a star graph.
Clearly, the vertex 0 is adjacent to all other elements of ZA .
Hence, the number of vertices adjacent with 0 is A Oe 1.
On the other hand, since a nonzero element v of ZA is adjacent to zero alone, the degree of v is 1.
n For the case with a different order of the ring, the degree of the elements can be seen below.
Theorem 1.
If eZA is the nilpotent graph of ZA where A = 2k for some k OO N, then dx = 12 A for x OO .
, 3, .
, 2k Oe .
and dy = A Oe 1 for y OO .
, 2, .
, 2k Oe .
Proof.
Suppose that S1 = .
, 3, .
, 2k Oe .
and S2 = .
, 2, .
, 2k Oe .
Note that S2 is the set of nilpotent elements of Zn so that all elements of S2 are adjacent to all elements of ZA .
On the other hand, the elements of S1 which are non-nilpotent only neighbour nilpotent elements.
So, the number of vertices adjacent to x OO S1 is dx = |S2 | = 21 A.
Since y OO S2 is adjacent to all of the other vertices except itself, dy = |S1 | |S2 | Oe 1 = A Oe 1.
n Based on the results obtained in previous studies, we formulate the Sombor energy of the given graph.
In this article, we explore the definition and properties of Sombor energy, building on previous work to offer a clearer understanding of its significance in graph theory.
MAIN RESULTS
Sombor Energy of Prime Number Order.
In 2021.
Gowtham and Swamy .
discovered another type of graph energy, the Sombor energy.
It was motivated by GutmanAos discovery of the Sombor index in 2021 [?].
The following is the definition of the Sombor matrix.
Definition 2.
The Sombor matrix of graph e is defined as matrix SO.
= .
oij ] with d2vi d2vj .
if vi and vj are adjacent .
where dvi is the degree of the vertex vi .
The Sombor matrix is used to find the eigenvalues to define the Sombor energy of a graph .
Definition 2.
, .
If SO.
is Sombor matrix of a graph e with eigenvalues 1 , 2 , .
A , then Sombor energy of the graph e is ESO .
= .
The size of the Sombor matrix is usually very large it is very difficult to calculate its determinant manually.
The lemma below will help us determine the determinant of a matrix that has a certain form.
Lemma 2.
Let r,s,t, u be real numbers and is a complex number, the determinant of the block matrix ( .
IA1 Oe rJA1 OeuJA2 yA1 OetJA1 yA2 ( .
IA2 Oe sJA2 of order A1 A2 can be expressed in the simplified form as ( .
A1 Oe1 ( .
A2 Oe1 [( Oe (A1 Oe .
( Oe (A2 Oe .
Oe A1 A2 t.
, .
where J is a rectangular matrix with all its entries 1 and I is the identity matrix.
From Definition 2.
1, we can determine the Sombor matrix and the characteristic polynomial of the nilpotent graph eZA .
Thus, it is necessary to find the determinant of SO.
ZA ).
Theorem 2.
The characteristic polynomial of the matrix SO.
ZA ) where A is a prime number is PeZA () = AOe2 .
Oe (A Oe .
(A2 Oe 2A .
Proof.
By Definition 2.
1 and Lemma 1.
1, we construct the Sombor matrix of the nilpotent graph of ZA with prime number order.
That is Oo A2 Oe 2A 2 A2 Oe 2A 2 A A A A2 Oe 2A 2 A2 Oe 2A 2 a Oo A2 Oe 2A 2 a SO.
ZA ) = .
Oo A2 Oe 2A 2 a Then we will find the eigenvalues of that matrix by using |SO.
ZA )OeI| = 0.
Since |SO.
Oe I| = |I Oe SO.
|, the determinant of SO.
Oe I is given by Oo Oo Oo Oe A2 Oe 2A 2 Oe A2 Oe 2A 2 A A A Oe A2 Oe 2A 2 Oo OeOoA2 Oe 2A 2 a Oe A2 Oe 2A 2 a Oo .
a Oe A2 Oe 2A 2 We can rewrite this matrix into I1 Oe A2 Oe 2A 2JAOe1y1 Oo Oo Oe A2 Oe 2A 2J1yAOe1 IAOe1 Using Lemma 2.
3, the characteristic polynomial of the matrix SO.
ZA ), where A is a prime number, is PeZA () =( .
1Oe1 ( .
AOe1Oe1 [( Oe .
Oe .
( Oe (A Oe 1 Oe .
Oe .
(A Oe .
(Oe A2 Oe 2A .
(Oe A2 Oe 2A .
] =AOe2 .
Oe (A Oe .
(A2 Oe 2A .
n Based on the Theorem 2.
4, we can easily formulate the general formula for Sombor energy.
Theorem 2.
The Sombor energy of eZA where A is a prime number is ESO .
ZA ) = 2 (A Oe .
(A2 Oe 2A .
Proof.
Based on Theorem 4, the eigenvalues of SO.
ZA ) are = 0 with multiplicity A Oe 2 and = A (A Oe .
(A2 Oe 2A .
each with multiplicity 1.
Hence, the Sombor energy of eZA where A is a prime number is ESO .
ZA ) =(A Oe .
| (A Oe .
(A2 Oe 2A .
| | Oe (A Oe .
(A2 Oe 2A .
| = (A Oe .
(A2 Oe 2A .
(A Oe .
(A2 Oe 2A .
=2 (A Oe .
(A2 Oe 2A .
n Sombor Energy when A is a Power of Two.
In this section, the Sombor energy for the order of squared prime numbers is The derivation involves constructing the associated matrix and determining its eigenvalues.
Theorem 2.
The characteristic polynomial of the matrix SO.
ZA ) where A = 2k for some k OO N is A Oo Oo A A2 PeZA () = 2 Oe1 ( (A Oe .
2 Oe1 2 Oe (A Oe .
Oe1 2 Oe A Oe 8A .
Proof.
By Definition 2.
1 and Lemma 1.
2, we construct the Sombor matrix of the nilpotent graph of ZA with the order a power of two.
That is Oo 0 Oo (A Oe .
2 A A A (A Oe .
Oo2 A A A (A Oe .
A A A (A Oe .
2 A A A (A Oe .
Oo2 (A Oe .
Oo2 A A A A A A SO.
ZA ) = a 0 0 A A A 0 a 0 0 A A A 0 a 0 0 a 0 Oo where = 21 5A2 Oe 8A 4.
Then we will find the eigenvalues of that matrix using |SO.
ZA ) Oe I| = 0.
Thus.
Oo Oe(A Oe .
Oe(A Oe .
|IOeSO.
ZA )| = Oe Oe(A Oe .
Oe(A Oe .
a a Oo Oe(A Oe .
Oo2 Oe(A Oe .
a a Oe a a a Oe a a a Oe a Oe a We can rewrite this matrix into Oo ( (A Oe .
I 2A Oe (A Oe .
2J 2A
Oe 12 5A2 Oe 8A 4J 2A
Oe 12 5A2 Oe 8A 4J 2A
I 2A
Using Lemma 2.
3, the characteristic polynomial of the matrix SO.
ZA ), where A = 2k for some k OO N is A A Oo A Oo PeZA () =( (A Oe .
2 Oe1 ( .
2 Oe1 Oe Oe 1 (A Oe .
2 Oe Oe1 0 5A Oe 8A 4 5A Oe 8A 4 Oo Oo A A2 = 2 Oe1 ( (A Oe .
2 Oe1 2 Oe (A Oe .
Oe1 2 Oe .
A Oe 8A .
n Building on the theorem above regarding the characteristic polynomial, we can readily derive the general formula for Sombor energy.
Theorem 2.
The Sombor energy of eZA where A = 2k for some k OO N is A A 2 A 2 Oo ESO .
ZA ) = Oe 1 (A Oe .
2 2(A Oe .
2 Oe 1 .
A2 Oe 8A .
Proof.
Based on Theorem 2.
6, the eigenvalues of the matrix SO.
ZA ) are = 0 with multiplicity 2 Oe 1, = Oe(A Oe .
2 with multiplicity 2A Oe 1, and = Oo Oe1 Oo 2 (A Oe .
Oe(A Oe .
Oe1 2 Oe 4 Oe .
A2 Oe 8A .
A A Oo
2 A 2
(A Oe .
Oe1 2 A 2(A Oe .
Oe 1 .
A Oe 8A .
A A each with multiplicity 1.
Hence, the Sombor energy of eZA with A = 2k for some k OO N is A A Oo ESO .
ZA ) = Oe 1 .
Oe 1 | Oe (A Oe .
2 A 2
(A Oe .
Oe1 2 2(A Oe .
2 Oe 1 .
A2 Oe 8A .
A Oo
2 A 2
(A Oe .
Oe1 2 Oe 2(A Oe .
Oe 1 .
A Oe 8A .
A A 2 A 2 Oo Oe 1 (A Oe .
2 2(A Oe .
2 Oe 1 .
A2 Oe 8A .
n In conclusion, we have highlighted the relationship between the graph representation of a ring and its Sombor energy.
By examining the properties of the graph, we have provided insights into how the graphAos structure influences its Sombor energy.
This work extends previous studies and contributes to a deeper understanding of the connection between algebraic structures and graph invariants.
Acknowledgement.
This research was conducted as part of the Merdeka Belajar Kampus Merdeka (MBKM) program.
The authors express their sincere gratitude to the Department of Mathematics at both the University of Mataram and the University of the Philippines Cebu for their support and collaboration, which significantly contributed to the completion of this study.
This work was financially supported by the 2025 International Collaboration Grant Scheme under the Non-Tax State Revenue (PNBP) program of University of Mataram.
References