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We translate some graph properties of ( R ) and Γ( R ) to some topological properties of Zariski topology. We prove that the facts “(1) The zero ideal of  R  is an anti fixed-place ideal. (2) Min( R ) does not have any isolated point. (3) Rad( ( R )) = 3. (4) Rad(Γ( R )) = 3. (5) Γ( R ) is triangulated (6)  ( R ) is triangulated.” are equivalent. Also, we show that if the zero ideal of a ring  R  is a fixed-place ideal, then dt t ( ( R )) = |ℬ( R )| and also if in addition |Min( R )| &gt; 2, then dt( ( R )) = |ℬ ( R )|. Finally, it is shown that dt( ( R )) is finite if and only if dt t ( ( R )) is finite if and only if Min( R ) is finite.

Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring