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≅M2(D) or D1×D2, for some division rings, D,D1 and D2. 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 P4. 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 Mn(D) for a division ring D and a natural number n>3 is not regular.