Topological Indices and Properties of the Prime Ideal Graph of a Commutative Ring and Its Line Graph

: Let 𝑅 be a commutative ring with identity, and 𝑃 be a prime ideal of 𝑅 . The prime ideal graph, denoted by Γ 𝑃 , is the graph where the set of vertices is 𝑅 ∖ {0} and two vertices are joined by an edge if their product belongs to 𝑃 . This paper will discuss topological indices and some properties of the prime ideal graph of a commutative ring and its line graph. Topological indices, such as the Wiener, first Zagreb, second Zagreb, Harary, Gutman, Schultz, and Harmonic indices, are related to the degree of vertices and the diameter of the graphs. In this paper, we also discuss the independence and domination numbers of the line graph.


Introduction
We use graphs in various disciplines, from electrical engineering and chemistry to computer science.In chemistry, one uses graph theory to solve molecular problems.Atoms and the bonds between atoms can be represented as the vertices and the edges of a graph, respectively.There are numerous applications of graph theory and group theory in chemistry, including using topological indices to represent the chemical structure with numerical values.Additionally, topological indices help predict molecular structures' chemical and physical properties.
In recent years, the representation of rings in graphs has been a topic of increasing research interest.One example is the commutative ring's zero-divisor graph.Given a commutative ring, its zero-divisor graph is a simple graph where each vertex represents an element in the ring, and two vertices x and y are joined by an edge if and only if xy is zero.Beck first introduced this concept in 1988 [17].Furthermore, in 2007, Anderson and Mulay obtained the zero-divisor graph's properties, such as its diameter being greater than or equal to 2 and its girth being greater than or equal to 4 [18].In 2021, Asir and Abikka proposed one of the topological indices for Zn, the Wiener index, for its zero-divisor graph [19].Alali et al. did the same, but on a broader scale by exploring topological indices and entropy using M-Polynomials [20].Readers are encouraged to delve into the study for a deeper comprehension of graphs of algebraic structures and their properties (see [21][22][23][24][25][26]).
A graph representing a commutative ring focusing on prime ideals was introduced by Salih and Jund in 2022.This graph is known as the prime ideal graph.One defines the prime ideal graph where the elements of the commutative ring, excluding the zero element, are represented as vertices in the graph.Two vertices x and y are joined by an edge if and only if xy belongs to a prime ideal.The obtained results include chromatic, clique, independence, and domination numbers [27].This study will discuss the prime ideal graph of a commutative ring and its line graph's properties, including the vertices' degree, diameter representing graph distance, domination number, independence number, and topological indices.The topological indices that will be investigated, particularly those related to vertex degree and distance, are the Wiener, first Zagreb, second Zagreb, Harary, Gutman, Schultz, and Harmonic indices.

Prime Ideal Graphs
This section discusses the prime ideal graph's properties.

Definition 2.1 [27]
For a commutative ring  and its prime ideal , the prime ideal graph Γ  is defined where the vertex set is  ∖ { 0 }, and two vertices  1 and  2 are adjacent whenever  1  2 ∈ .In this paper, we will discuss the results for any finite commutative ring R with a prime ideal P.
Next, we provide a lemma regarding the degree of a vertex in Γ  .

Lemma 2.1
In   , the degree of a vertex is given by: Below are several definitions regarding graph properties, namely the clique, domination, independence, and chromatic numbers, as well as the results of previous research on prime ideal graphs.

Theorem 2.1 [27]
The diameter of   is less than or equal to 2.

Definition 2.3 [28]
A dominating set is a subset  of (Γ) such that every vertex in (Γ) ∖  is joined by an edge to some vertex in .The domination number, (Γ), is the order of a minimum dominating set.

Figure 2 . 2 .
Figure 2.2.Γ  with  = {1 + , 1 +  2 ,  +  2 }.Any finite commutative ring is, in particular Artinian, and commutative Artinian rings  are finite products ∏ of Artinian local rings.An example of Artinian local ring is provided by ℤ 2 [] ⟨ 3 +1⟩ , another examples are group algebras of finite Abelian -groups over fields of characteristic .If  is a finite commutative -group and  is a field of characteristic , then  is a local ring which is commutative.

Proof.Lemma 2 . 2
If  ∈  * , then  is joined by an edge to all  ∈  ∖ {0, } .Consequently, deg() =  − 2, for every  ∈  * .If  ∈  ∖ , then  is only adjacent to the vertices in  except 0. Consequently, deg() =  − 1, for every  ∈  ∖ .Now, let's discuss the distance in graphs.Below is a lemma about distance in the prime ideal graph.The distance of between two vertices x and y in   is given by Proof.If ,  ∈  ∖ , it means that ,  ∉  * , indicating that x is not joined by an edge to y.However, we may find  ∈  * such that  ∈  * and  ∈  * , resulting in a path between  and  passing through the vertex  ∈  * .Consequently, the distance between  and  is 2. If  ∈  * , then  is joined by edge to all vertices  ∈  * since  ∈  * .Thus, (, ) = 1.

1 .Theorem 2 . 3 Definition 2 . 7 [ 30 ]Theorem 2 . 4 Definition 2 . 8 [ 30 ]Definition 2 . 9 [ 31 ]Theorem 2 . 6 Theorem 2 . 7 Definition 3 . 12 [ 34 ]Theorem 2 . 9 3 .Example 3 . 1
The domination number of   , (  ) = 1, 2. The independence number of   , (  ) =  − , Now, we move to the next topic of discussion, namely topological indices.Some of the topological indices to be discussed include the first Zagreb, second Zagreb, Wiener, Harary, Harmonic, Gutman, and Schultz indices.Below, we provide definitions and results of these topological indices.Definition 2.6 [29] For a connected graph Γ, the sum of pairs of vertices in Γ is called the Wiener index of Γ, The Wiener index of the prime ideal graph is Proof.The distance between vertices are given as follows 1.If ,  ∈  * , then the distance between x and y is 1, and there are ( −1 2 ) pairs of vertices.2. If ,  ∈ \, then the distance between x and y is 2, and there are ( − 2 ) pairs of vertices.3.If  ∈  * and  ∈ \, then the distance between  and  is 1, and there are ( − 1)( − ) pairs of vertices.Based on the above cases, For a connected graph Γ, the sum of squares of the degrees of each vertex in Γ is called the first Zagreb index of Γ, The first Zagreb index of the prime ideal graph is given by Proof.Based on Lemma 3.1, we have the following: For a connected graph Γ, the sum of the product of degrees of each pair of vertices joint by an edge in Γ is called the second Zagreb index of Γ, Theorem 2.5 The second Zagreb index of the prime ideal graph is given by Proof.Based on Lemma 3.1 1.If ,  ∈  * , then () = () =  − 2 and there are ( − 2 ) pairs of vertices.2. If  ∈  * and  ∈ \, then () =  − 2 and deg() =  − 1, and there are ( − 1)( − ) pairs of vertices.Hence, For a connected graph Γ, the sum of the inverses of the distances between all pairs of vertices in Γ is called the Harary index of Γ, The Harary index of the prime ideal graph is Proof.Using the same approach as the Wiener index, we obtain Definition 2.10 [32] For a connected graph Γ, the sum of the product of the degree of each pair of vertices and the distance between vertices, is called the Gutman index of Γ, The Gutman index of the prime ideal graph is Proof.Based on the distance between vertices, and degree of the vertices, we have the following.Definition 2.11 [33] For a connected graph Γ, the sum of the product between the distance between vertices and the sum of the degrees of each pair of vertices is called the Schultz index of Γ Theorem 2.8 The Schultz index of the prime ideal graph is Proof.Based on the distance between vertices, and degree of the vertices, as in the Gutman index, we obtain the following.For a connected graph Γ, the sum of two divided the sum of the degrees of each pair of vertices joint by an edge, is called the Harmonic index of Γ The Harmonic index of the prime ideal graph is Proof.Based on the degree of vertices, following the same approach as the second Zagreb index, we obtain the following: The Line Graph of the Prime Ideal Graph of a Commutative Ring This section discusses the line graph of the prime ideal graph of a commutative ring.Definition 3.1 [28] The line graph of a graph Γ, L(Γ), is a graph such that 1.Each vertex of L(Γ) is an edge of Γ. 2. Two vertices of L(Γ) are joined by an edge if and only if their corresponding edges in Γ have a common endpoint.Let  =  6 with  = {0,2,4}.Then, we have (Γ  ) as follows

Theorem 3 . 1
If L(Γ P ) as the line graph of prime ideal graph of a commutative ring, then diam(L(Γ P )) ≤ 2.