  For a set of non-negative integers $L$, the $L$-intersection number of a
graph is the smallest number $l$ for which there is an assignment on the
vertices to subsets $A_v \subseteq \{1,\dots, l\}$, such that every two
vertices $u,v$ are adjacent if and only if $|A_u \cap A_v|\in L$. The bipartite
$L$-intersection number is defined similarly when the conditions are considered
only for the vertices in different parts. In this paper, some lower bounds for
the (bipartite) $L$-intersection number of a graph for various types $L$ in
terms of the minimum rank of graph are obtained.
