Some lower bounds for the L-intersection number of graphs

‎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 of subsets Av⊆{1,…‎,‎l} to vertices v‎, ‎such that every two vertices u,v are adjacent if and only if |Au∩Av|∈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. ‎To achieve the main results we employ the inclusion matrices of set systems and show that how the linear algebra techniques give elegant proof and stronger results in some cases.