total perfect codes in graphs realized by commutative rings

Let r be a commutative ring with unity not equal to zero and let γ(r) be a zero-divisor graph realized by r. for a simple, undirected, connected graph g = (v, e), a total perfect code denoted by c(g) in g is a subset c(g) ⊆ v (g) such that |n(v) ∩ c(g)| = 1 for all v ∈ v (g), where n(v) denotes the open neighbourhood of a vertex v in g. in this paper, we study total perfect codes in graphs which are realized as zero-divisor graphs. we show a zero-divisor graph realized by a local commutative ring with unity admits a total perfect code if and only if the graph has degree one vertices. we also show that if γ(r) is a regular graph on |z ^∗ (r)| number of vertices, then r is a reduced ring and |z ^∗ (r)| ≡ 0(mod 2), where z ^∗ (r) is a set of non-zero zero-divisors of r. we provide a characterization for all commutative rings with unity of which the realized zero-divisor graphs admit total perfect codes. finally, we determine the cardinality of a total perfect code in γ(r) and discuss the significance of the study of total perfect codes in graphs realized by commutative rings with unity.