Let V be a left R-module where R is a (not necessarily commutative)
ring with unit. The intersection graph \cG(V) of proper R-submodules of V
is an undirected graph without loops and multiple edges defined as follows: the
vertex set is the set of all proper R-submodules of V, and there is an edge
between two distinct vertices U and W if and only if U∩W=0. We
study these graphs to relate the combinatorial properties of \cG(V) to the
algebraic properties of the R-module V. We study connectedness, domination,
finiteness, coloring, and planarity for \cG (V). For instance, we find the
domination number of \cG (V). We also find the chromatic number of \cG(V)
in some cases. Furthermore, we study cycles in \cG(V), and complete subgraphs
in \cG (V) determining the structure of V for which \cG(V) is planar