The Rank of a Cograph - Royle, Gordon F. (The Electronic Journal of Combinatorics, 2003)
The rank of the adjacency matrix of a graph is bounded above by the number of distinct non-zero rows of that matrix. In general, the rank is lower than this number because there may be some non-trivial linear combination of the rows equal to zero. We show the somewhat surprising result that this never occurs for the class of cographs. Therefore, the rank of a cograph is equal to the number of distinct non-zero rows of its adjacency matrix.