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 ).

We investigate the structural properties of a graph defined on the ring. The Adjacency between two different vertices and is determined by the bilinear congruence. We analyze three fundamental cases, and for distinct odd primes. We describe the graph's breakdown into unit and non-unit vertex subsets. The unit subgraph forms disjoint cliques, with sizes depending on Euler's totient function. In contrast, the zero-divisor subgraph shows more complex behaviour governed by annihilation ideals. We establish general properties, including degree formulas, determination of maximum clique sizes in each component, determining the diameter, computing the girth, locating the graph centers, and finding the measures of vertex and edge connectivity. Additionally, we characterize independent sets and prove the existence of Hamiltonian cycles and supereulerian properties under certain connectivity conditions. Our results show how the prime factorization of influences these properties.

For a composite number 𝒏, we explore a topological space on the ring of integers modulo 𝒏, we use basis sets formed by solutions to quadratic residue equations, where elements are units modulo 𝒏. This definition allows us to investigate the algebraic relationships among solutions, focusing on properties like clopen sets, closure, and interior operations. Additionally, we examine the continuity of mappings between the group of units in ℤ/𝒏ℤ and quadratic residues, alongside the quotient topology induced on this group of units. A comparison of the separation properties of these topological spaces is also presented

In this work,we investigate Frobenius companion matrices with entries from the ring of integers modulo a prime number. We introduce a mapping 𝝍to construct a directed graph, where vertices represent these matrices and edges are defined by 𝝍. Our analysis focuses on the structure and properties of this directed graph, including vertex degrees, cycles, and connected components, in relation to theeigenvalues of the matrices

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®.