  The intersection ideal graph $\Gamma(S)$ of a semigroup $S$ is a simple
undirected graph whose vertices are all nontrivial left ideals of $S$ and two
distinct left ideals $I, J$ are adjacent if and only if their intersection is
nontrivial. In this paper, we investigate the connectedness of $\Gamma(S)$. We
show that if $\Gamma(S)$ is connected then $diam(\Gamma(S)) \leq 2$. Further we
classify the semigroups such that the diameter of their intersection graph is
two. Other graph invariants, namely perfectness, planarity, girth, dominance
number, clique number, independence number etc. are also discussed. Finally, if
$S$ is union of $n$ minimal left ideals then we obtain the automorphism group
of $\Gamma(S)$.
