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

We translate some graph properties of AG(R) and Gamma(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(AG(R)) = 3. (4) Rad(Gamma(R)) = 3. (5) Gamma(R) is triangulated (6) AG(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)(AG(R)) = vertical bar B(R)vertical bar and also if in addition vertical bar Min(R)vertical bar > 2, then dt(AG(R)) = vertical bar B(R)vertical bar. Finally, it is shown that dt(AG(R)) is finite if and only if dt(t)(AG(R)) is finite if and only if Min(R) is finite.