Some topological indices of the prime graph of commutative ring zpq

Introduction

Graph theory has become a very well-known and rapidly growing field of mathematics due to its extensive theoretical advances and diverse applications to real-world problems. Although graph theory is still a relatively new field of research, it has yielded many profound and novel discoveries over the past 20 years. In biological, social, physical, and information systems, diagrams can be used to represent a variety of relationships and processes. A topological indices is a real number assigned to the graph whose structure must be a constant. Topological indexes, sometimes is called molecular structure descriptors. More topological indices depend on degrees. The Weiner index, also known as the Wiener index or the 1 st topological index, was indeed introduced by Harry Wiener in (1947). This index is a simple and widely used measure of the molecular topological structure in chemistrical. Sylvester introduced the term graph in an 1878 article in Nature Publishing [1] . The 1 st book in graph theory was published by Denes in 1936 [2] . In the last decade of the 19th century, algebraic graph theory began to develop rapidly, and a large number of research articles were published in this branch of graph theory. The fascinating field of algebraic graph theory studies how algebra and graph theory interact. Algebraic methods can be used to prove graph theory facts in surprising and elegant ways. There are many interesting algebraic objects related to graphs. In recent years, the study of algebraic graph theory has become increasingly important. Algebraic graph theory involves converting properties of graphs into algebraic properties and then using algebraic results and techniques to derive theorems about graphs. However, many algebraic topics can also be understood by converting them into graphs and exploiting the properties of graphs. In (1999), Anderson and Livingston [3] assigned a simple graph to each commutative ring R and examined to interaction between the ring-theoretical conditions and the graph-theoretic properties of. Akbari et al. In (2010) Bhavanari et al. [3] , they considered binding rings R (not necessarily commutative) and defined a new concept "primary graph of R" (denoted by PG(R)). They presented some examples and obtained some basic important results related to PG(R). (2013) Chelvam and Asir. [5] Studied the advantages in commutative ring ensemble graphs. In (2014), Patra and Kalita [6] studied prime graphs of commutative rings. In (2023), Nermen J. Khalel andNabeel E. Arif [7] studied Associate graph of a commutative ring.

In the articles, we study more topological metrics that need degrees of examples: eccentricity index [8] , connectivity index [9] , sum connectivity index [10] , 1 st and 2 nd index [11] , forgetting indicator [12] , exponential geometric arithmetic [13] , atomic bond connectivity index [13] and harmonic index [13] . Topological indexing of symmetry group graphs was introduced in (2019) Abdussakir [14] , also examined in (2020) G. R. Roshini [15] , and also in (2020) G. R. Roshini [16], Alaa.J and Akram.S [17] in (2021) study topological indices and (Schultz and Hosoya) polynomials of the intersection graph of subgroup of the group 𝑍𝑟.

Primary Results Definition 2.1:

Let 𝑍 𝑝𝑞 a ring. A prime graph of ring𝑍 𝑝 , indicated as PG (𝑧 𝑝𝑞 ) is a graph where PG(𝑧 𝑝𝑞 ) (V, E) where V(PG(𝑧 𝑝𝑞 ))= 𝑍 𝑝𝑞 and E(PG(𝑧 𝑝𝑞 ))={𝑎𝑏 ̅̅̅ , 𝑎. 𝑏 = 0 𝑎𝑛𝑑 𝑎 ≠ 𝑏}.

Remark2.2: Let PG (𝑧 𝑝𝑞 ) be a graph of 𝑅 with𝑅 = 𝑍 𝑝𝑞 . Then 1. (0) and ( 1) adjacent to all vertices.. 2. A distance between any two distinct vertices a and b is 𝑑(𝑎, 𝑏) ≤ 2.

Corollary2.3:

If 𝑍 𝑝𝑞 be a ring, 𝑝, 𝑞 is a prime number, then PG (𝑧 𝑝𝑞 ) is a 1) Connected 2) 𝛼(PG(𝑧 𝑝𝑞 ) = 1

Methods and Materials

In this paper, all diagrams are simplfy, bounded, connect, and UN directional. For the graph 𝐺 = (𝑉 (𝐺), 𝐸 (𝐺)) let 𝑑𝑒𝑔(𝑢) be the degree of vertex 𝑢 in 𝐺. If 𝑑𝑒𝑔(𝑢) = 0, then u is an isolated vertex. Let 𝑑 (𝑢, 𝑣) be the distance between two distinct vertices u and v. The 𝑒𝑐𝑐(𝑢) of the vertex 𝑢 is 𝑒𝑐𝑐(𝑢) = 𝑠𝑢𝑝 {𝑑(𝑢, 𝑣): 𝑣 ∈ 𝑉(𝐺)} Next definitions refers to a topological indices of a graph 𝐺 Eccentricity index of 𝐺 is [8] ξ 𝑐 (𝐺))) =∑

The Eccentricityty connectivity index of 𝐺 is [9]

The Sum connectivity index of 𝐺 is [10] 𝑆(G) = ∑ 1 √deg(𝑢) + deg (𝑣)

A 1 st zagreb index of G is [11] 𝑀 1 (G) = ∑ (deg(𝑢)) 2

A 2 nd zagreb index of 𝐺 is [11] 𝑀 2 (G) = ∑ deg(𝑢) . deg (𝑣)

The forgotten index of 𝐺 is [12] 𝐹(G) = ∑ (deg( 𝑢)) 3

The atomic bond connectivity index of G is [13] 𝐴𝑏𝑐

The Geometric arithmetic index of 𝐺 is [13] 𝐺𝐴(G) = ∑ 2 √deg ( 𝑢) . deg (𝑣)

Harmonic index of 𝐺 is [13] 𝐻 + (𝑝𝑞 -𝑝 -𝑞 + 1) + √ pq + q -3 q(pq -1)

Conclusions

In this study, formulas for some degree and eccentricity based topological indices are proposed for prime graph of ring 𝑍𝑝𝑞, where 𝑝, 𝑞 is a prime number. To investigate further, examine the non-zero graph of the ring 𝑍𝑝𝑞, where 𝑝, 𝑞 are prime numbers

References

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