On the inclusion ideal graph of a ring

The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all non-trivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ⊆ J or J ⊆ I . In this paper, we show that In(R) is not connected if and only if R ≅ M 2 ( D ) or D 1 × D 2 , for some division rings, D , D 1 and D 2 . Moreover, if R is connected, then diam ( In ( R ) ) ⩽ 3 . We prove that if In(R) is a tree, then In(R) is a star graph or P 4 . Also, In(R) is a complete graph if and only if R is a uniserial ring. Next, it is shown that the inclusion ideal graph of M n ( D ) for a division ring D and a natural number n > 3 is not regular.