The rings considered in this paper are finite commutative rings with identity, which are not fields. For any ring [Formula: see text] which is not a field and which is not necessarily finite, we denote the set of all zero-divisors of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. Let [Formula: see text] denote the zero-divisor graph of [Formula: see text] and for a finite ring [Formula: see text], let [Formula: see text] denote the maximum degree of [Formula: see text]. We denote [Formula: see text] by [Formula: see text]. The aim of this paper is to study some properties of [Formula: see text].

In our study we introduced soft i-open sets and soft i-star-generalized-w-closed sets in soft bi-topological spaces, , using the notion of soft i-open sets in soft-topological-space, . We besides that give examples to clarify these relationships while presenting some essential characteristics and relationships between various groups of sets. Besides that we studied the property of soft bi-topologically extended and non-soft bi-topologically extended of soft i-open sets in soft bi-topological spaces by proofs and examples.

Let Γ be a nontrivial connected graph, c : V Γ ⟶ ℕ be a vertex colouring of Γ , and L i be the colouring classes that resulted, where i = 1,2 , … , k . A metric colour code for a vertex a of a graph Γ is c a = d a , L 1 , d a , L 2 , … , d a , L n , where d a , L i is the minimum distance between vertex a and vertex b in L i . If c a ≠ c b , for any adjacent vertices a and b of Γ , then c is called a metric colouring of Γ as well as the smallest number k satisfies this definition which is said to be the metric chromatic number of a graph Γ and symbolized μ Γ . In this work, we investigated a metric colouring of a graph Γ Z n and found the metric chromatic number of this graph, where Γ Z n is the zero-divisor graph of ring Z n .

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