The aim of this paper is to represent a bitopological representation (𝑉,𝜏𝑆1,𝜏𝑆2) of divisor graph 𝐺=(𝑉,𝐸) defining in a finite commutative rings in which every vertex 𝑣 is adjacent with a vertex 𝑢 if and only if 𝑔.𝑐.𝑑(𝑢,𝑣)=1. Then some properties of this bitopological space were investigated.

This paper introduces a finite topological space 𝜏𝑝 on the group of units modulo a prime 𝑝, defined by its basis of conjugate residue pairs {𝛼, 𝑝 − 𝛼} for all units 𝛼 ∈ 𝑈𝑝. We investigate the fundamental topological concepts such as point-set topology, separation axioms, and characterise the structure and behaviour of this topology. Additionally, we examine a function 𝑓 from 𝜏𝑝 to the topology of quadratic residues 𝜏𝑄, mapping each unit to its square modulo 𝑝. We analyse the continuity, openness of 𝑓, and explore its implications for separation properties. Furthermore, we define a quotient topology on 𝑈𝑝 based on the equivalence relation 𝑥 ∼ 𝑦 if and only if 𝑥2 ≡ 𝑦2 𝑚𝑜𝑑 𝑝, showing that the resulting quotient space is homeomorphic to (𝑄𝑝 , 𝜏𝑄 ).

The aim of this paper is to prove that the Diophantine exponent 𝐷𝑖𝑜( 𝑑−𝜷(𝑙𝛽)) is bounded by 𝑙𝑜𝑔𝑀(𝛽)𝑙𝑜𝑔𝛽+(𝑑−1)𝑙𝑜𝑔|𝑎𝑑|+1𝑙𝑜𝑔𝛽, where 𝛽 is an algebraic number, and 𝑀(𝛽) is the Mahler measure of 𝛽. In addition, a new result will be founded about transcendental number 𝑎 when 𝐷𝑖𝑜(𝑎) is infinite.

In this work, we investigate essential definitions, defining 𝐺 as a simple graph with vertices in ℤ𝑛 and subgraphs Γ𝑢 and Γ𝑞 as unit residue and quadratic residue graphs modulo 𝑛, respectively. The investigation extends to the degree of 𝐺, Γ𝑢, and Γ𝑞, illuminating the properties of these subgraphs in the context of quadratic congruences.

In the present article, we are going to highlight the relation between different digraphs (cycles) of finite commutative ring Zn for a natural number n, under the map (a,b) → (a+b,ab). The algorithm, which is used to perform the calculations, has been built in MATLAB®.