1. INTRODUCTIONKnowing the π-electron energy of conjugated carbon molecules in chemistry that had figured out using H¨uckel theory, aligns perfectly with the concept of energy as defined in the realm of graph theory. This remarkable coincidence provides a special significance to results related to graph energy. The H¨uckel theory in quantum chemistry provides a method to calculate the energy levels of π-electrons in conjugated carbon molecules [1]. When these energy calculations are paralleled with graph energy, a fascinating intersection of chemistry and mathematics emerges.
Similarly, in the study of algebraic structures, graph theory ofers a powerful framework for understanding the properties of groups [2]. The power graph, for instance, provides valuable insights into the algebraic structure of the group [3] or on the ring [4]. It reflects the order of elements, and subgroup relationships, and can highlight symmetries and other structural properties. Power graphs are often simpler than other associated graphs, such as the Cayley or commuting graph, making them useful for studying finite groups. The power graph of a group G is a graph whose vertex set consists of the elements of the group, and two vertices are adjacent if one is a power of the other [5]. Building on this, we derive the general formulas of various graph energies for the power graph on cyclic group. Some scholars have examined the intersection power graph of groups, determining that the graph is Eulerian when the group’s order is even [6]. Others have contributed findings on the rainbow connection number for the power graph of finite groups [7]. Additionally, researchers have identified the automorphism groups for various power graphs of finite groups [8], and recently, there has been a growing interest in the study of graph energy for the non-coprime graph associated with the dihedral group[9] and on integer modulo group [10].
Recently, Putra et. al. have published the Sombor Energy of the nilpotent graph of the integer modulo group, which is also one of the cyclic groups [11]. In this article, we investigate not just Sombor energy but including degree sum energy, degree exponential energy, degree subtraction energy, maximum degree adjacency energy, degree square sum energy, Sombor energy, and Randi´c energy on the cyclic group in general, especially for the prime power order. We also identified a correlation between the graph spectrum and the corresponding energy values obtained.
2. Energy of GraphIn this section, we aim to calculate the characteristic polynomial and graph energy of the power graph of cyclic groups with prime power order. These calculations include degree sum energy, degree exponential energy, degree subtraction energy, maximum degree adjacency energy, degree square sum energy, Sombor energy, and Randi´c energy. Below is the definition of the energy of a graph.
Definition 2.1.[1]Let Γ be a graph and η be an eigenvalue of the matrix graph of Γ, then the energy of Γ is defined as
Finding the determinant of a small matrix is very straightforward, but not for a large matrix size. This determinant is very important for finding the eigenvalues of a matrix. Below is a property that can help us to find the determinant of a square matrix of order σ.
Lemma 2.2.[12]If we have a square matrix with order σ, where and ν are real numbers, then
2.1. Degree Sum Energy.In this section, we explore the concept of degree sum energy, examining its definition, properties, and significance within the context of our study. We analyze how degree sum energy is calculated, its implications for the power graph, and its role in understanding the underlying algebraic structure.
Definition 2.3. [13]Let Γ be a graph with E as a set of its edges, then the matrix of degree sum defined as with,
where are the degree of vertices and , respectively.
In the following theorem, we calculate the characteristic polynomial of the degree-sum matrix for the power graph of a cyclic group of order , denoted as , where σ is a power of prime.
Theorem 2.4.The characteristic polynomial of with where p is a prime number and is
Proof. Suppose that , with where p is a prime number and For any can be written as with . Thus, we have two cases:
Case 1, Wlog, let , and . So, there exist such that implies . Thus, b is a power of a.
Case 2, If and σ are not relatively prime, then , for some . Wlog, let , so there exists a natural number q such that . Then Hence, a and b are adjacent in both cases.
Because all two diferent elements of are neighbours, the power graphs of must be a complete graph with order σ. So, the degree of all vertices is
Then by Definition 2.3 we get the degree sum matrix of is
Therefore, the characteristic polynomial of is found by calculating which gives
By using Lemma 2.2, we get
Theorem 2.5.The degree sum energy of with where p is a prime number and is
Proof. By Theorem 2.4, we obtain the eigenvalues of the degree sum matrix of by solving which gives
Thus, with multiplicity and with multiplicity 1. Using Definition 2.1 which states that the degree sum energy of is the sum of all its absolute eigenvalues, we find that the degree sum energy of is
2.2. Degree Exponent Energy.In this section, we explore the concept of the characteristic polynomial and the degree exponent energy in greater detail. We examine how the characteristic polynomial is derived and its properties. Additionally, we discuss the degree exponent energy, focusing on its significance and its role in understanding the behavior of certain functions or systems.
Definition 2.6.[14]Let Γ be a graph, then the matrix of degree exponent of Γ is with
where are the degree of vertices and , respectively.
Theorem 2.7.The characteristic polynomial of with where is a prime number and is
Proof. Based on the proof of Theorem 2.4, we know that with is a prime power has degrees. By Definition 2.6, we obtain the degree exponent matrix of the graph by raising its degrees to the powers of the other vertices’ degrees. Thus,
By calculating , we obtain the characteristic polynomial of the degree exponent matrix of , namely
Theorem 2.8.The degree exponent energy of with where p is a prime number and is
Proof. By solving from Theorem 2.7, we obtain
The multiplicities are and 1, respectively. Therefore, the degree exponent energy of is
2.3. Degree Subtraction Energy.This section explores the characteristic polynomial and the degree subtraction energy of the nilpotent graph of . First, we discuss how the characteristic polynomial is derived and its mathematical properties. The characteristic polynomial helps calculate the eigenvalues of a matrix, which is used to compute the degree of subtraction energy.
Definition 2.9.[15]Let Γ be a graph, the degree subtraction matrix of Γ iswith
where are the degree of vertices and , respectively.
Theorem 2.10.The characteristic polynomial of with where p is a prime number and is
Proof. By Definition 2.9, we can form the degree subtraction matrix of the graph with σ is a prime power by subtracting the degree of each vertex. Since all vertices of have the same degree, . Therefore, , and , where O is the zero matrix. It is easy to check that □
Theorem 2.11. The degree subtraction energy of with where is a prime number and is 0.
Proof. By Theorem 2.10, the eigenvalues of are all 0. Therefore, the degree subtraction energy of is 0. □
2.4. Maximum Degree Adjacency.This section explores the characteristic polynomial and the maximum degree energy. The maximum degree matrix is based on the maximum degree of two distinct vertices. The characteristic polynomial is crucial for calculating the eigenvalues of a matrix. Next, we compute the maximum degree energy, which is the summation of absolute eigenvalues of the maximum degree matrix.
Definition 2.12.[16]The maximum degree matrix of graph Γ with E is a set of its edges defined with the maximum degree of vertex of graph Γ which is with
Theorem 2.13.The characteristic polynomial of with where p is a prime number and is (9)
Proof. Since with σ is a prime power is a complete graph, all of its vertices have the same degree. By the proof of Theorem 2.4 , the degree of each vertex is . Therefore, max . Thus, we obtain the maximum degree adjacency matrix of below:
Thus, the characteristic polynomial of the matrix is
Theorem 2.14.The maximum degree energy of graph with where p is a prime number and is
Proof. By Theorem 2.13, the eigenvalues of the matrix are with multiplicity and with multiplicity 1. Therefore, by the definition of energy, the energy of the maximum degree matrix of is
2.5. Degree Square Sum.The degree square sum matrix was defined similarly to the degree sum matrix. However, before we summed the vertex’s degrees, each degree value was squared. By calculating its characteristic polynomial, we determine its eigenvalues to compute the degree square sum energy of the nilpotent graph of
Definition 2.15.[17]The degree square sum matrix of graph Γ is defined as with
Theorem 2.16.The characteristic polynomial of with where is a prime number and is
Proof. By the definition of the degree square sum matrix, we can form the matrix by adding the squares of the degrees of two distinct vertices of with is a prime power. Thus,
Thus, the characteristic polynomial of matrix is
Theorem 2.17.The degree square sum of graph with where is a prime number and is
Proof. By Theorem 2.16, the eigenvalues of are with multiplicity and with multiplicity 1. Therefore, the degree square sum energy of is
2.6. Sombor Energy.The Sombor energy is the summation of absolute eigenvalues of the Sombor matrix. In this section, we examine the characteristic polynomial to get the eigenvalues of the Sombor matrix to calculate its Sombor energy. We begin with the Sombor matrix, defined as the square root of the sum of the squares of the degrees of two adjacent vertices on the graph.
Definition 2.18. [18]Let Γ be a graph, defined Sombor matrix of graph Γ with,
Theorem 2.19.Let is power graph of with where p is a prime number and . Then the characteristic polynomial of is
Proof. Based on Definition 2.18, we obtain the Sombor matrix of with is a prime power, which is
Thus, the characteristic polynomial of is
Theorem 2.20.The Sombor energy with where p is a prime number and is equal to
Proof. By Theorem 2.19, the eigenvalues of are with multiplicity and with multiplicity 1. Therefore, we obtain
2.7. Randi´c Energy.Randi´c matrix is defined as the reciprocal of the square root of the product of two degrees. In this section, we calculate the characteristic polynomial and Randi´c energy by summing the absolute eigenvalues of the Randi´c matrix.
Definition 2.21.[19]Let Γ be a graph, defined Randi´c matrix of graph Γ with,
Theorem 2.22. The characteristic polynomial of with where p is a prime number and is
Proof. Based on Definition 2.21 and the proof of Theorem 2.4, we can form the Randi´c energy of with is a prime power, represented by
Hence, we obtain
Theorem 2.23.The Randi´c energy of with where p is a prime number and is equal to 2.
Proof. Based on Theorem 2.22, the eigenvalues of are with multiplicity and with multiplicity 1. Therefore,
Here, we introduce the graph’s spectrum and relate it to its energy, showing how spectral characteristics reveal the graph’s total energy.
Definition 2.24.Let Γ be a graph with σ vertices, be eigenvalues of matrix of graph Γ with each multiplicity of its eigenvalues is then spectral of Γ is
Spectral radius of graph Γ defined as
From the definition given above and all of the previous results, we obtain the following consequence.
Corollary 2.25.The degree sum energy, degree exponent energy, degree subtraction energy, maximum degree energy, degree square sum energy, Sombor energy and Randi´c energy of graph with where p is a prime number and is two times its spectral radius.
3. ConclusionBased on Section 2, the characteristic polynomial of the degree sum matrix, degree exponential matrix, degree subtraction matrix, maximum degree adjacency matrix, degree square sum matrix, Sombor matrix, and Randi´c matrix of nilpotent graph of where σ is a power of prime number respectively are:
(1)
(2)
(3)
(4)
(6)
(7)
From the characteristic polynomial of each matrix, we calculate degree sum energy, degree exponential energy, degree subtraction energy, maximum degree adjacency energy, degree square sum energy, Sombor energy, and Randi´c energy, respectively namely:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
The last one is the degree sum energy, degree exponent energy, degree subtraction energy, maximum degree energy, degree square sum energy, Sombor energy and Randi´c energy of graph with where is a prime number and is two times its spectral radius.
These results are crucial for initiating research on the energy of the graph, particularly for other cyclic groups or possibly for finite groups.
CRediT