A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graphG with weighted edges and no loops. A graph associated with a n◊n non‐singular matrix with zero entries on the diagonal such that all its (n 1)◊ (n 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one‐ and two‐vertex deleted subgraphs of a NSSD. It is shown that a necessary and sucient condition for two‐vertex deleted subgraphs of G and of the graph ( G 1 ) associated with G 1 to remain NSSDs is that the submatrices belonging to them, derived from G and G 1 , are inverses. Moreover, an algorithm yielding what we term plain NSSDs is presented. This algorithm can be used to determine if a graph G with a terminal vertex is not a NSSD.