The Metric Chromatic Number of Zero Divisor Graph of a Ring Z[sub.n]

Let Γ be a nontrivial connected graph, c:V(Γ )⟶ℕ be a vertex colouring of Γ , and L[sub.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[sub.1]),d(a,L[sub.2]),…,d(a,L[sub.n])), where d(a,Li) is the minimum distance between vertex a and vertex b in L[sub.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[sub.n]) and found the metric chromatic number of this graph, where Γ (Z[sub.n]) is the zero-divisor graph of ring Z[sub.n].