Independence Number and Covering Vertex Number of Γ(R(+)M)

Some Properties of the Zero-Divisor Graphs of Idealization Ring R(+)M

“…Al-Labadi M in [10], characterized the zero-divisor graph of idealization ring when R is an integral domain.…”

“…(1) If ann(M) � 0, then by [10] we have d((0, 1), (r i , t))1 and d((r i , t 1 ), (r j , t 2 )) � 2 for all r i , r j ∈ ann(M). Hence, the wiener index of the graph…”

“…(1) If ann(M) � 0, then by [10], we have R � Z 2 i.e, Γ(Z 2 (+)Z 2 ) � (0, 1) { } is an empty graph which is not a Hamiltonian graph.…”

“…Recently, Allabadi [9,10], studied the properties of the zero-divisors graph of idealization ring such as when the zero-divisors graph of idealization ring is Planar graph, divisor graph and Eulerian graph and the independence number of the zero-divisor graph of the idealization ring.…”

“…(2) If ann(M) ≠ 0, then by [10] we have Γ(R(+)M) � (0, { t), (r, m): r ∈ ann(M) and t, m ∈ M}. So, we can not find the cycle between all vertices in the graph Γ(R(+)M) that is not a Hamiltonian graph.…”