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.